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A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
(Formerly M1413 N0552)
632

%I M1413 N0552

%S 0,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,80782,195025,

%T 470832,1136689,2744210,6625109,15994428,38613965,93222358,225058681,

%U 543339720,1311738121,3166815962,7645370045,18457556052,44560482149,107578520350,259717522849

%N Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).

%C Sometimes also called lambda numbers.

%C Also denominators of continued fraction convergents to sqrt(2): 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/A000129.

%C Number of lattice paths from (0,0) to the line x=n-1 consisting of U=(1,1), D=(1,-1) and H=(2,0) steps (i.e., left factors of Grand Schroeder paths); for example, a(3)=5, counting the paths H, UD, UU, DU and DD. - _Emeric Deutsch_, Oct 27 2002

%C a(2*n) with b(2*n) := A001333(2*n), n >= 1, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = +1 (see Emerson reference). a(2*n+1) with b(2*n+1) := A001333(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = -1.

%C Bisection: a(2*n+1) = T(2*n+1, sqrt(2))/sqrt(2) = A001653(n), n >= 0 and a(2*n) = 2*S(n-1,6) = 2*A001109(n), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310. - _Wolfdieter Lang_, Jan 10 2003

%C Consider the mapping f(a/b) = (a + 2b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 3/2, 7/5, 17/12, 41/29, ... converging to 2^(1/2). Sequence contains the denominators. - _Amarnath Murthy_, Mar 22 2003

%C This is also the Horadam sequence (0,1,1,2). Lim_{n->infinity} a(n)/a(n-1) = sqrt(2) + 1. - _Ross La Haye_, Aug 18 2003

%C Number of 132-avoiding two-stack sortable permutations.

%C For n > 0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 3. - _Herbert Kociemba_, Jun 02 2004

%C Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 1, s(n) = 2. - _Herbert Kociemba_, Jun 02 2004

%C Counts walks of length n from a vertex of a triangle to another vertex to which a loop has been added. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004

%C Apart from initial terms, Pisot sequence P(2,5). See A008776 for definition of Pisot sequences. - _David W. Wilson_

%C Sums of antidiagonals of A038207 [Pascal's triangle squared]. - _Ross La Haye_, Oct 28 2004

%C The Pell primality test is "If N is an odd prime, then P(N)-Kronecker(2,N) is divisible by N". "Most" composite numbers fail this test, so it makes a useful pseudoprimality test. The odd composite numbers which are Pell pseudoprimes (i.e., that pass the above test) are in A099011. - _Jack Brennen_, Nov 13 2004

%C a(n) = sum of n-th row of triangle in A008288 = A094706(n) + A000079(n). - _Reinhard Zumkeller_, Dec 03 2004

%C Pell trapezoids (cf. A084158); for n > 0, A001109(n) = (a(n-1) + a(n+1))*a(n)/2; e.g., 1189 = (12+70)*29/2. - _Charlie Marion_, Apr 01 2006

%C (0!a(1), 1!a(2), 2!a(3), 3!a(4), ...) and (1,-2,-2,0,0,0,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - _Tom Copeland_, Oct 29 2007

%C Let C = (sqrt(2)+1) = 2.414213562..., then for n > 1, C^n = a(n)*(1/C) + a(n+1). Example: C^3 = 14.0710678... = 5*(0.414213562...) + 12. Let X = the 2 X 2 matrix [0, 1; 1, 2]; then X^n * [1, 0] = [a(n-1), a(n); a(n), a(n+1)]. a(n) = numerator of n-th convergent to (sqrt(2)-1) = 0.41421356... = [2, 2, 2, ...], the convergents being [1/2, 2/5, 5/12, ...]. - _Gary W. Adamson_, Dec 21 2007

%C A = sqrt(2) = 2/2 + 2/5 + 2/(5*29) + 2/(29*169) + 2/(169*985) + ...; B = ((5/2) - sqrt(2)) = 2/2 + 2/(2*12) + 2/(12*70) + 2/(70*408) + 2/(408*2378) + ...; A+B = 5/2. C = 1/2 = 2/(1*5) + 2/(2*12) + 2/(5*29) + 2/(12*70) + 2/(29*169) + ... - _Gary W. Adamson_, Mar 16 2008

%C Prime Pell numbers with an odd index gives the RMS value (A141812) of prime RMS numbers (A140480). - _Ctibor O. Zizka_, Aug 13 2008

%C From _Clark Kimberling_, Aug 27 2008: (Start)

%C Related convergents (numerator/denominator):

%C lower principal convergents: A002315/A001653

%C upper principal convergents: A001541/A001542

%C intermediate convergents: A052542/A001333

%C lower intermediate convergents: A005319/A001541

%C upper intermediate convergents: A075870/A002315

%C principal and intermediate convergents: A143607/A002965

%C lower principal and intermediate convergents: A143608/A079496

%C upper principal and intermediate convergents: A143609/A084068. (End)

%C Equals row sums of triangle A143808 starting with offset 1. - _Gary W. Adamson_, Sep 01 2008

%C Binomial transform of the sequence:= 0,1,0,2,0,4,0,8,0,16,..., powers of 2 alternating with zeros. - _Philippe Deléham_, Oct 28 2008

%C a(n) is also the sum of the n-th row of the triangle formed by starting with the top two rows of Pascal's triangle and then each next row has a 1 at both ends and the interior values are the sum of the three numbers in the triangle above that position. - Patrick Costello (pat.costello(AT)eku.edu), Dec 07 2008

%C Starting with offset 1 = eigensequence of triangle A135387 (an infinite lower triangular matrix with (2,2,2,...) in the main diagonal and (1,1,1,...) in the subdiagonal. - _Gary W. Adamson_, Dec 29 2008

%C Starting with offset 1 = row sums of triangle A153345. - _Gary W. Adamson_, Dec 24 2008

%C From _Charlie Marion_, Jan 07 2009: (Start)

%C In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:

%C let a(k,0) = 1, a(k,1) = 2k; for n > 0, a(k,2n) = 2*a(k,2n-1) + a(k,2n-2)

%C and a(k,2n+1) = (2k)*a(k,2n) + a(k,2n-1);

%C let b(k,0) = 1, b(k,1) = 2k+1; for n > 0, b(k,2n) = 2*b(k,2n-1) + b(k,2n-2)

%C and b(k,2n+1) = (2k)*b(k,2n) + b(k,2n-1).

%C For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5, 17/12, 41/29.

%C In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then

%C k*a(k,2n)^2 - a(k,2n-1)*a(k,2n+1) = k = k*a(k,2n-2)*a(k,2n) - a(k,2n-1)^2 and

%C b(k,2n-1)*b(k,2n+1) - k*b(k,2n)^2 = k+1 = b(k,2n-1)^2 - k*b(k,2n-2)*b(k,2n);

%C for example, if k=1 and n=3, then a(1,n) = a(n+1) and

%C 1*a(1,6)^2 - a(1,5)*a(1,7) = 1*169^2 - 70*408 = 1;

%C 1*a(1,4)*a(1,6) - a(1,5)^2 = 1*29*169 - 70^2 = 1;

%C b(1,5)*b(1,7) - 1*b(1,6)^2 = 99*577 - 1*239^2 = 2;

%C b(1,5)^2 - 1*b(1,4)*b(1,6) = 99^2 - 1*41*239 = 2.

%C Cf. A001333, A142238, A142239, A153313, A153314, A153315, A153316, A153317, A153318.

%C (End)

%C Starting with offset 1 = row sums of triangle A155002, equivalent to the statement that the Fibonacci series convolved with the Pell series prefaced with a "1": (1, 1, 2, 5, 12, 29, ...) = (1, 2, 5, 12, 29, ...). -_Gary W. Adamson_, Jan 18 2009

%C It appears that P(p) == 8^((p-1/2)) mod p, p = prime; analogous to [Schroeder, p.90]: Fp == 5^((p-1)/2)) mod p. Example: Given P(11) = 5741, == 8^5 mod 11. Given P(17) = 11336689, == 8^8 mod 17 since 17 divides (8^8 - P(l7)). - _Gary W. Adamson_, Feb 21 2009

%C Equals eigensequence of triangle A154325. - _Gary W. Adamson_, Feb 12 2009

%C Another combinatorial interpretation of a(n-1) arises from a simple tiling scenario. Namely, a(n-1) gives the number of ways of tiling a 1 X n rectangle with indistinguishable 1 X 2 rectangles and 1 X 1 squares that come in two varieties, say, A and B. For example, with C representing the 1 X 2 rectangle, we obtain a(4)=12 from AAA, AAB, ABA, BAA, ABB, BAB, BBA, BBB, AC, BC, CA and CB. - _Martin Griffiths_, Apr 25 2009

%C a(n+1) = 2*a(n) + a(n-1), a(1)=1, a(2)=2 was used by Theon from Smyrna. - _Sture Sjöstedt_, May 29 2009

%C The n-th Pell number counts the perfect matchings of the edge-labeled graph C_2 x P_(n-1), or equivalently, the number of domino tilings of a 2 x (n-1) cylindrical grid. - Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009

%C Number of units of a(n) belongs to a periodic sequence: 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1. - _Mohamed Bouhamida_, Sep 04 2009

%C As a fraction: 1/79 = 0.0126582278481... or 1/9799 = 0.000102051229...(1/119 and 1/10199 for sequence in reverse). - _Mark Dols_, May 18 2010

%C Lim_{n->infinity} (a(n)/a(n-1) - a(n-1)/a(n)) tends to 2.0. Example: a(7)/a(6) - a(6)/a(7) = 169/70 - 70/169 = 2.0000845... - _Gary W. Adamson_, Jul 16 2010

%C Numbers n such that 2*n^2+-1 is a square. - _Vincenzo Librandi_, Jul 18 2010

%C Starting (1, 2, 5, ...) = INVERTi transform of A006190: (1, 3, 10, 33, 109, ...). - _Gary W. Adamson_, Aug 06 2010

%C [u,v] = [a(n), a(n-1)] generates all Pythagorean triples [u^2-v^2, 2uv, u^2+v^2] whose legs differ by 1. - _James R. Buddenhagen_, Aug 14 2010

%C An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 21 and 336, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A078057. - _Johannes W. Meijer_, Aug 15 2010

%C Let the 2 X 2 square matrix A=[2, 1; 1, 0] then a(n) = the (1,1) element of A^(n-1). - _Carmine Suriano_, Jan 14 2011

%C Define a t-circle to be a first-quadrant circle tangent to the x- and y-axes. Such a circle has coordinates equal to its radius. Let C(0) be the t-circle with radius 1. Then for n > 0, define C(n) to be the next larger t-circle which is tangent to C(n - 1). C(n) has radius A001333(2n) + a(2n)*sqrt(2) and each of the coordinates of its point of intersection with C(n + 1) is a(2n + 1) + (A001333(2n + 1)*sqrt(2))/2. See similar Comments for A001109 and A001653, Sep 14 2005. - _Charlie Marion_, Jan 18 2012

%C A001333 and A000129 give the diagonal numbers described by Theon from Smyrna. - _Sture Sjöstedt_, Oct 20 2012

%C Pell numbers could also be called "silver Fibonacci numbers", since, for n >= 1, F(n+1) = ceiling(phi*F(n)), if n is even and F(n+1) = floor(phi*F(n)), if n is odd, where phi is the golden ratio, while a(n+1) = ceiling(delta*a(n)), if n is even and a(n+1) = floor(delta*a(n)), if n is odd, where delta = delta_S = 1+sqrt(2) is the silver ratio. - _Vladimir Shevelev_, Feb 22 2013

%C a(n) is the number of compositions (ordered partitions) of n-1 into two sorts of 1's and one sort of 2's. Example: the a(3)=5 compositions of 3-1=2 are 1+1, 1+1', 1'+1, 1'+1', and 2. - _Bob Selcoe_, Jun 21 2013

%C Between every two consecutive squares of a 1 X n array there is a flap that can be folded over one of the two squares. Two flaps can be lowered over the same square in 2 ways, depending on which one is on top. The n-th Pell number counts the ways n-1 flaps can be lowered. For example, a sideway representation for the case n = 3 squares and 2 flaps is \\., .//, \./, ./_., ._\., where . is an empty square. - _Jean M. Morales_, Sep 18 2013

%C Define a(-n) to be a(n) for n odd and -a(n) for n even. Then a(n) = A005319(k)*(a(n-2k+1) - a(n-2k)) + a(n-4k) = A075870(k)*(a(n-2k+2) - a(n-2k+1)) - a(n-4k+2)). - _Charlie Marion_, Nov 26 2013

%C An alternative formulation of the combinatorial tiling interpretation listed above: Except for n=0, a(n-1) is the number of ways of partial tiling a 1 X n board with 1 X 1 squares and 1 X 2 dominoes. - _Matthew Lehman_, Dec 25 2013

%C Define a(-n) to be a(n) for n odd and -a(n) for n even. Then a(n) = A077444(k)*a(n-2k+1) + a(n-4k+2). This formula generalizes the formula used to define this sequence. - _Charlie Marion_, Jan 30 2014

%C a(n-1) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 1, 1; 1, 1, 1; 0, 1, 1], [0, 1, 1; 0, 1, 1; 1, 1, 1], [0, 1, 0; 1, 1, 1; 1, 1, 1] or [0, 0, 1; 1, 1, 1; 1, 1, 1]. - _R. J. Mathar_, Feb 03 2014

%C a(n+1) counts closed walks on K2 containing two loops on the other vertex. Equivalently the (1,1) entry of A^(n+1) where the adjacency matrix of digraph is A=(0,1;1,2). - _David Neil McGrath_, Oct 28 2014

%C For n >= 1, a(n) equals the number of ternary words of length n-1 avoiding runs of zeros of odd lengths. - _Milan Janjic_, Jan 28 2015

%C This is a divisibility sequence (i.e., if n|m then a(n)|a(m)). - _Tom Edgar_, Jan 28 2015

%C A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - _Michael Somos_, Jan 03 2017

%C a(n) is the number of compositions (ordered partitions) of n-1 into two kinds of parts, n and n', when the order of the 1 does not matter, or equivalently, when the order of the 1' does not matter. Example: When the order of the 1 does not matter, the a(3)=5 compositions of 3-1=2 are 1+1, 1+1'=1+1, 1'+1', 2 and 2'. (Contrast with entry from _Bob Selcoe_ dated Jun 21 2013.) - _Gregory L. Simay_, Sep 07 2017

%C Number of weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n and for which {1,...,n} has exactly one minimal element for the weak ordering R. - _J. Devillet_, Sep 28 2017

%C Also the number of matchings in the (n-1)-centipede graph. - _Eric W. Weisstein_, Sep 30 2017

%C Let A(r,n) be the total number of ordered arrangements of an n+r tiling of r red squares and white tiles of total length n, where the individual tile lengths can range from 1 to n. A(r,0) corresponds to a tiling of r red squares only, and so A(r,0)=1. Let A_1(r,n) = Sum_{j=0..n} A(r,j) and let A_s(r,n) = Sum_{j=0..n} A_(s-1)(r,j). Then A_0(1,n) + A_2(3,n-4) + A_4(5,n-8) + ... + A_(2j) (2j+1, n-4j) = a(n) without the initial 0. - _Gregory L. Simay_, May 25 2018

%C (1, 2, 5, 12, 29, ...) is the fourth INVERT transform of (1, -2, 5, -12, 29, ...), as shown in A073133. - _Gary W. Adamson_, Jul 17 2019

%D P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.

%D A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.

%D S.-M. Belcastro, Tilings of 2 x n Grids on Surfaces, preprint.

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 941.

%D J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 53.

%D John Derbyshire, Prime Obsession, Joseph Henry Press, 2004, see p. 16.

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.1.

%D Shaun Giberson and Thomas J. Osler, Extending Theon's Ladder to Any Square Root, Problem 3858, Elementa, No. 4 1996.

%D R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149.

%D Thomas Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.

%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

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%D Manfred R. Schroeder, "Number Theory in Science and Communication", 5th ed., Springer-Verlag, 2009, p. 90.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. J. A. Sloane and Simone Sandri, <a href="/A000129/b000129.txt">Table of n, a(n) for n = 0..1000</a> (First 500 terms from N. J. A. Sloane).

%H M. Abrate, S. Barbero, U. Cerruti, N. Murru, <a href="http://dx.doi.org/10.1155/2013/543913">Construction and composition of rooted trees via descent functions</a>, Algebra, Volume 2013 (2013), Article ID 543913, 11 pages.

%H Paraskevas K. Alvanos, Konstantinos A. Draziotis, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Draziotis/draz2.html">Integer Solutions of the Equation y^2 = Ax^4 + B</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.4.

%H Tewodros Amdeberhan, <a href="http://www.math.temple.edu/~tewodros/solutions/solu.html">Solution to problem #10663 (AMM)</a>

%H Ayoub B. Ayoub, <a href="http://www.jstor.org/stable/27646417">Fibonacci-like sequences and Pell equations</a>, The College Mathematics Journal, Vol. 38 (2007), pp. 49-53.

%H Ovidiu Bagdasar, Eve Hedderwick, Ioan-Lucian Popa, <a href="https://doi.org/10.1016/j.endm.2018.05.011">On the ratios and geometric boundaries of complex Horadam sequences</a>, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 63-70.

%H Aseem R. Baranwal, Jeffrey Shallit, <a href="https://arxiv.org/abs/1902.00503">Critical exponent of infinite balanced words via the Pell number system</a>, arXiv:1902.00503 [cs.FL], 2019.

%H Elena Barcucci, Antonio Bernini, Renzo Pinzani, <a href="http://ceur-ws.org/Vol-2113/paper8.pdf">A Gray code for a regular language</a>, Semantic Sensor Networks Workshop 2018, CEUR Workshop Proceedings (2018) Vol. 2113.

%H J.-L. Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.

%H Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, <a href="http://math.colgate.edu/~integers/t46/t46.Abstract.html">Motzkin paths with a restricted first return decomposition</a>, Integers (2019) Vol. 19, A46.

%H M. Barnabei, F. Bonetti and M. Silimbani, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p25">Two permutation classes related to the Bubble Sort operator</a>, Electronic Journal of Combinatorics 19(3) (2012), #P25.

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%H J. Bodeen, S. Butler, T. Kim, X. Sun, S. Wang, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p7">Tiling a strip with triangles</a>, El. J. Combinat. 21 (1) (2014) P1.7.

%H Latham Boyle, Paul J. Steinhardt, <a href="https://arxiv.org/abs/1608.08220">Self-Similar One-Dimensional Quasilattices</a>, arXiv preprint arXiv:1608.08220 [math-ph], 2016.

%H B. Bradie, <a href="https://projecteuclid.org/euclid.mjms/1312232719">Extensions and Refinements of some properties of sums involving Pell Numbers</a>, Miss. J. Math. Sci 22 (1) (2010) 37-43.

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H Geoffrey B. Campbell, Aleksander Zujev, <a href="http://arxiv.org/abs/1511.07424">Gaussian integer solutions for the fifth power taxicab number problem</a>, arXiv:1511.07424 [math.NT], 2015.

%H Frédéric Chapoton, <a href="https://arxiv.org/abs/1809.00575">A note on gamma triangles and local gamma vectors</a>, arXiv:1809.00575 [math.CO], 2018.

%H C. O. Chow, S. M. Ma, T. Mansour, and M. Shattuck, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from43to54.pdf">Counting permutations by cyclic peaks and valleys</a>, Annales Mathematicae et Informaticae, (2014), Vol. 43, pp. 43-54.

%H M. Couceiro, J. Devillet, and J.-L. Marichal, <a href="http://arxiv.org/abs/1709.09162">Quasitrivial semigroups: characterizations and enumerations</a>, arXiv:1709.09162 [math.RA], 2017.

%H Phan Thuan Do, Thi Thu Huong Tran, Vincent Vajnovszki, <a href="https://arxiv.org/abs/1809.00742">Exhaustive generation for permutations avoiding a (colored) regular sets of patterns</a>, arXiv:1809.00742 [cs.DM], 2018.

%H C. M. da Fonseca, <a href="https://doi.org/10.1016/j.amc.2014.03.064">Unifying some Pell and Fibonacci identities</a>, Applied Mathematics and Computation, Volume 236, Jun 01 2014, Pages 41-42.

%H Mahadi Ddamulira, <a href="https://arxiv.org/abs/1906.06330">On the x-coordinates of Pell equations which are products of two Pell numbers</a>, arXiv:1906.06330 [math.NT], 2019.

%H E. Deutsch, <a href="https://www.jstor.org/stable/2589194">A formula for the Pell numbers, Problem 10663</a>, Amer. Math. Monthly 107 (No. 4, 2000), solutions pp. 370-371.

%H E. S. Egge and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205206">132-avoiding two-stack sortable permutations, Fibonacci numbers, and Pell numbers</a>, arXiv:math/0205206 [math.CO], 2002.

%H Shalosh B. Ekhad and Tewodros Amdeberhan, <a href="https://www.math.temple.edu/~tewodros/solutions/10663.PDF">Solution to Problem #10663</a>.

%H C. Elsner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Elsner/elsner15.html">On Error Sums for Square Roots of Positive Integers with Applications to Lucas and Pell Numbers</a>, J. Int. Seq. 17 (2014) # 14.4.4.

%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242, Ex. 1, pp. 237-238.

%H S. Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, 2014, 5, 2226-2234.

%H Sergio Falcon, <a href="http://ijism.org/administrator/components/com_jresearch/files/publications/IJISM_816_FINAL.pdf">Generating Function of Some k-Fibonacci and k-Lucas Sequences</a>, International Journal of Innovation in Science and Mathematics (2019) Vol. 7, Issue 2, 2347-9051.

%H Sergio Falcón, <a href="http://doi.org/10.26713/cma.v10i3.1221">Binomial Transform of the Generalized k-Fibonacci Numbers</a>, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.

%H Bakir Farhi, <a href="https://www.emis.de/journals/JIS/VOL22/Farhi/farhi19.html">Summation of Certain Infinite Lucas-Related Series</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.

%H M. C. Firengiz, A. Dil, <a href="http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf">Generalized Euler-Seidel method for second order recurrence relations</a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.

%H Felix Flicker, <a href="https://arxiv.org/abs/1707.09371">Time quasilattices in dissipative dynamical systems</a>, arXiv:1707.09371 [nlin.CD], 2017. Also <a href="http://doi.org/10.21468/SciPostPhys.5.1.001">SciPost</a> Phys. 5, 001 (2018).

%H Shaun Giberson and Thomas J. Osler, <a href="https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/extending-theons-ladder-to-any-square-root">Extending Theon's Ladder to Any Square Root</a>, College Mathematics Journal, May, 2004.

%H Juan B. Gil, Aaron Worley, <a href="https://arxiv.org/abs/1901.02619">Generalized metallic means</a>, arXiv:1901.02619 [math.NT], 2019.

%H Taras Goy, <a href="http://dx.doi.org/10.30755/NSJOM.08406">Pell numbers identities from Toeplitz-Hessenberg determinants</a>, Novi Sad J. Math., 49 (2) (2019), 87-94.

%H Martin Griffiths, <a href="http://dx.doi.org/10.1080/0020739X.2013.790512">Pell identities via a quadratic field</a>, International Journal of Mathematical Education in Science and Technology, 2013.

%H R. P. Grimaldi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Grimaldi/grimaldi5.html">Tilings, Compositions, and Generalizations</a>, J. Int. Seq. 13 (2010), 10.6.5.

%H M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, <a href="http://www.jstor.org/stable/2968551">Problem 47</a>, Amer. Math. Monthly, 4 (1897), 25-28.

%H R. J. Hetherington, <a href="/A000129/a000129.pdf">Letter to N. J. A. Sloane, Oct 26 1974</a>

%H Gábor Hetyei, <a href="https://math.uncc.edu/sites/math.uncc.edu/files/fields/preprint_archive/paper/2019_16.pdf">The type B permutohedron and the poset of intervals as a Tchebyshev transform</a>, University of North Carolina-Charlotte (2019).

%H Nick Hobson, <a href="/A000129/a000129.py.txt">Python program for this sequence</a>

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special Properties of the Sequence W(n){a,b; p,q}</a>, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/9-3/horadam-a.pdf">Pell Identities</a>, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252, 263.

%H Haruo Hosoya, <a href="http://www.hyle.org/journal/issues/19-1/hosoya.htm">What Can Mathematical Chemistry Contribute to the Development of Mathematics?</a>, HYLE--International Journal for Philosophy of Chemistry, Vol. 19, No.1 (2013), pp. 87-105.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=135">Encyclopedia of Combinatorial Structures 135</a>

%H M. Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, J. Int. Seq. 18 (2015), #15.4.7.

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997) 122-126.

%H C. J. Kirchen, <a href="/A000129/a000129_1.pdf">Letter to N. J. A. Sloane, Feb 11 1974</a>

%H Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16.

%H K. Kuhapatanakul, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kuhapatanakul/kuha4.html">On the Sums of Reciprocal Generalized Fibonacci Numbers</a>, J. Int. Seq. 16 (2013) #13.7.1.

%H Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate, <a href="https://arxiv.org/abs/1904.13002">The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d))</a>, arXiv:1904.13002 [math.NT], 2019.

%H Shirley Law, <a href="http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/viewFile/dmAT0159/4536">Hopf Algebra of Sashes</a>, in FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 621-632.

%H H. Li, T. MacHenry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/MacHenry/machenry7.html">Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences</a>, J. Int. Seq. 16 (2013) #13.3.5, example 46.

%H Edouard Lucas, <a href="http://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf">The Theory of Simply Periodic Numerical Functions</a>, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

%H T. Mansour, M. Shattuck, <a href="http://dx.doi.org/10.1007/s12044-014-0166-7">A statistic on n-color compositions and related sequences</a>, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 2, May 2014, pp. 127-140.

%H A. Moghaddamfar, H. Tajbakhsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Moghaddamfar/moghadam3.html">More Determinant Representations for Sequences</a>, Journal of Integer Sequences, 17 (2014), #14.5.6.

%H Sophie Morier-Genoud, Valentin Ovsienko, <a href="https://arxiv.org/abs/1812.00170">Farey boat II. Q-deformations: q-deformed rationals and q-continued fractions</a>, arXiv:1812.00170 [math.CO], 2018.

%H Sophie Morier-Genoud, Valentin Ovsienko, <a href="https://hal.archives-ouvertes.fr/hal-02270545/">q-deformed rationals and q-continued fractions</a>, (2019) [math].

%H Emanuele Munarini, <a href="https://doi.org/10.1515/puma-2015-0028">A generalization of André-Jeannin's symmetric identity</a>, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.

%H Mariana Nagy, Simon R. Cowell, Valeriu Beiu, <a href="https://arxiv.org/abs/1902.05944">Survey of Cubic Fibonacci Identities - When Cuboids Carry Weight</a>, arXiv:1902.05944 [math.HO], 2019.

%H Ahmet Öteleş, <a href="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/ASUO/mathematics/anale2019vol2/06_oteles.pdf">Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers</a>, An. Şt. Univ. Ovidius Constantą, (2019) Vol. 27, Issue 2, 109-120.

%H Ahmet Öteleş, Zekeriya Y. Karata, Diyar O. Mustafa Zangana, <a href="https://www.emis.de/journals/JIS/VOL21/Oteles/ote5.html">Jacobsthal Numbers and Associated Hessenberg Matrices</a>, J. Int. Seq., 21 (2018), #18.2.5.

%H Arzu Özkoç, <a href="http://dx.doi.org/10.1186/s13662-015-0486-7">Some algebraic identities on quadra Fibona-Pell integer sequence</a>, Advances in Difference Equations, 2015, 2015:148.

%H Hao Pan, <a href="https://doi.org/10.1016/j.disc.2006.03.067">Arithmetic properties of q-Fibonacci numbers and q-Pell numbers</a>, Discr. Math., 306 (2006), 2118-2127.

%H D. Panario, M. Sahin, Q. Wang, <a href="http://www.emis.de/journals/INTEGERS/papers/n78/n78.Abstract.html">A family of Fibonacci-like conditional sequences</a>, INTEGERS, Vol. 13, 2013, #A78.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de Séries Génératrices et Quelques Conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Raul Prisacariu, <a href="https://scholarworks.umt.edu/tme/vol15/iss3/7/">Generating k-Pell Infinite Series Using Whittaker's Formula</a>, The Mathematics Enthusiast: Vol. 15 : No. 3 , Article 7, 2018.

%H C. Raissi and J. Pei, <a href="http://www.cs.sfu.ca/~jpei/publications/SeqPatternBound-KDD11.pdf">Towards Bounding Sequential Patterns</a>, KDD'11, Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, 2011.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1801.00466">An approximate Jerusalem square whose side equals a Pell number</a>, arXiv:1801.00466 [math.CO], 2018.

%H José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.

%H John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>

%H Michelle Rudolph-Lilith, <a href="http://arxiv.org/abs/1508.07894">On the Product Representation of Number Sequences, with Application to the Fibonacci Family</a>, arXiv preprint arXiv:1508.07894 [math.NT], 2015.

%H J. L. Schiffman, <a href="http://archives.math.utk.edu/ICTCM/i/24/C027.html">Exploring the Fibonacci sequence of order two with CAS technology</a>, Electronic Proceedings of the Twenty-fourth Annual International Conference on Technology in Collegiate Mathematics, Orlando, Florida, March 22-25, 2012, Paper C027.

%H Jon E. Schoenfield, <a href="/A000129/a000129.txt">Prime factorization of a(n) for n = 1..300</a>

%H James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Sellers/sellers4.html">Domino Tilings and Products of Fibonacci and Pell Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.

%H Mark A. Shattuck, <a href="http://www.emis.de/journals/INTEGERS/papers/j5/j5.pdf">Tiling proofs of some formulas for the Pell numbers of odd index</a>, Integers, 9 (2009), 53-64.

%H M. Shattuck, <a href="https://www.emis.de/journals/JIS/VOL17/Shattuck/shattuck8.html">Combinatorial Proofs of Some Formulas for Triangular Tilings</a>, Journal of Integer Sequences, 17 (2014), #14.5.5.

%H Nanci Smith, <a href="http://www.fq.math.ca/Scanned/4-4/elementary4-4.pdf">Problem B-82 An Integer Valued Function</a>, Fib. Quart., 4 (1966), 374-375.

%H Yüksel Soykan, <a href="https://doi.org/10.9734/AJARR/2019/v6i130144">On Generalized Third-Order Pell Numbers</a>, Asian Journal of Advanced Research and Reports (2019) Vol. 6, No. 1, Article No. AJARR.51635, 1-18.

%H Yüksel Soykan, <a href="https://doi.org/10.9734/AIR/2019/v20i230154">On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers</a>, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article No. AIR.51824.

%H Yüksel Soykan, Mehmet Gümüş, Melih Göcen, <a href="https://doi.org/10.13140/RG.2.2.21008.97289">A Study On Dual Hyperbolic Generalized Pell Numbers</a>, Zonguldak Bülent Ecevit University (Zonguldak, Turkey, 2019).

%H Robin James Spivey, <a href="https://doi.org/10.7546/nntdm.2019.25.3.170-184">Close encounters of the golden and silver ratios</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 170-184.

%H R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Moments, Narayana numbers and the cut and paste for lattice paths</a>

%H Ping Sun, <a href="http://arxiv.org/abs/1506.07256">Enumeration of standard Young tableaux of shifted strips with constant width</a>, arXiv:1506.07256 [math.CO], 24 Jun 2015.

%H Gy. Tasi and F. Mizukami, <a href="http://dx.doi.org/10.1023/A:1019163812482">Quantum algebraic-combinatoric study of the conformational properties of n-alkanes</a>, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63).

%H A. Tekcan, M. Tayat, M. E. Ozbek, <a href="http://dx.doi.org/10.1155/2014/897834">The diophantine equation 8x^2-y^2+8x(1+t)+(2t+1)^2=0 and t-balancing numbers</a>, ISRN Combinatorics, Volume 2014, Article ID 897834, 5 pages.

%H Ian Walker, <a href="http://www.mth.pdx.edu/~caughman/Ian%20Walker%20501.pdf">Explorations in Recursion with John Pell and the Pell Sequence</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentipedeGraph.html">Centipede Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellNumber.html">Pell Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellPolynomial.html">Pell Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagorassConstant.html">Pythagoras's Constant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareRoot.html">Square Root</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareTriangularNumber.html">Square Triangular Number</a>

%H Meral Yasar and Durmus Bozkurt, <a href="https://doi.org/10.1016/j.amc.2011.11.089">Another proof of Pell identities by using the determinant of tridiagonal matrix</a>, Appl. Math. Comput., 218 (2012), pp. 6067-6071.

%H Leon Zaporski, Felix Flicker, <a href="https://arxiv.org/abs/1811.00331">Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences</a>, arXiv:1811.00331 [nlin.CD], 2018.

%H Abdelmoumène Zekiri, Farid Bencherif, Rachid Boumahdi, <a href="https://www.emis.de/journals/JIS/VOL21/Zekiri/zekiri4.html">Generalization of an Identity of Apostol</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.

%H Jianqiang Zhao, <a href="http://arxiv.org/abs/1507.04917">Finite Multiple zeta Values and Finite Euler Sums</a>, arXiv preprint arXiv:1507.04917 [math.NT], 2015.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).

%F G.f.: x/(1 - 2*x - x^2). - _Simon Plouffe_ in his 1992 dissertation.

%F G.f.: Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (2*k + x)/(1 + 2*k*x) ) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (x + 1 + k)/(1 + k*x) ) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (x + 3 - k)/(1 - k*x) ) may all be proved using telescoping series. - _Peter Bala_, Jan 04 2015

%F a(n) = 2*a(n-1) + a(n-2), a(0)=0, a(1)=1.

%F a(n) = ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/(2*sqrt(2)).

%F For initial values a(0) and a(1), a(n) = ((a(0)*sqrt(2)+a(1)-a(0))*(1+sqrt(2))^n + ((a(0)*sqrt(2)-a(1)+a(0))*(1-sqrt(2))^n)/(2*sqrt(2)). - _Shahreer Al Hossain_, Aug 18 2019

%F a(n) = integer nearest a(n-1)/(sqrt(2) - 1), where a(0) = 1. - _Clark Kimberling_

%F a(n) = Sum_{i, j, k >= 0: i+j+2k=n} (i+j+k)!/(i!*j!*k!).

%F a(n)^2 + a(n+1)^2 = a(2n+1) (1999 Putnam examination).

%F a(2n) = 2*a(n)*A001333(n). - _John McNamara_, Oct 30 2002

%F a(n) = ((-i)^(n-1))*S(n-1, 2*i), with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(-2, x)= -1.

%F Binomial transform of expansion of sinh(sqrt(2)x)/sqrt(2). E.g.f.: exp(x)sinh(sqrt(2)x)/sqrt(2). - _Paul Barry_, May 09 2003

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k+1)2^k. - _Paul Barry_, May 13 2003

%F a(n-2) + a(n) = (1 + sqrt(2))^(n-1) + (1 - sqrt(2))^(n-1) = A002203(n-1). (A002203(n))^2 - 8(a(n))^2 = 4(-1)^n. - _Gary W. Adamson_, Jun 15 2003

%F Unreduced g.f.: x(1+x)/(1 - x - 3x^2 - x^3); a(n) = a(n-1) + 3a(n-2) + a(n-2). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004

%F a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)2^(n-2k). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004

%F Apart from initial terms, inverse binomial transform of A052955. - _Paul Barry_, May 23 2004

%F a(n)^2 + a(n+2k+1)^2 = A001653(k)*A001653(n+k); e.g., 5^2 + 70^2 = 5*985. - _Charlie Marion_ Aug 03 2005

%F a(n+1) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))2^k/2. - _Paul Barry_, Aug 28 2005

%F a(n) = a(n-1) + A001333(n-1) = A001333(n) - a(n-1) = A001109(n)/A001333(n) = sqrt(A001110(n)/A001333(n)^2) = ceiling(sqrt(A001108(n)/2)). - _Henry Bottomley_, Apr 18 2000

%F a(n) = F(n, 2), the n-th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006

%F Define c(2n) = -A001108(n), c(2n+1) = -A001108(n+1) and d(2n) = d(2n+1) = A001652(n); then ((-1)^n)*(c(n) + d(n)) = a(n). [Proof given by _Max Alekseyev_.] - _Creighton Dement_, Jul 21 2005

%F a(r+s) = a(r)*a(s+1) + a(r-1)*a(s). - _Lekraj Beedassy_, Sep 03 2006

%F a(n) = (b(n+1) + b(n-1))/n where {b(n)} is the sequence A006645. - _Sergio Falcon_, Nov 22 2006

%F From _Miklos Kristof_, Mar 19 2007: (Start)

%F Let F(n) = a(n) = Pell numbers, L(n) = A002203 = companion Pell numbers (A002203):

%F For a >= b and odd b, F(a+b) + F(a-b) = L(a)*F(b).

%F For a >= b and even b, F(a+b) + F(a-b) = F(a)*L(b).

%F For a >= b and odd b, F(a+b) - F(a-b) = F(a)*L(b).

%F For a >= b and even b, F(a+b) - F(a-b) = L(a)*F(b).

%F F(n+m) + (-1)^m*F(n-m) = F(n)*L(m).

%F F(n+m) - (-1)^m*F(n-m) = L(n)*F(m).

%F F(n+m+k) + (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = F(n)*L(m)*L(k).

%F F(n+m+k) - (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = L(n)*L(m)*F(k).

%F F(n+m+k) + (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = L(n)*F(m)*L(k).

%F F(n+m+k) - (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = 8*F(n)*F(m)*F(k). (End)

%F a(n+1)*a(n) = 2*Sum_{k=0..n} a(k)^2 (a similar relation holds for A001333). - _Creighton Dement_, Aug 28 2007

%F a(n+1) = Sum_{k=0..n} binomial(n+1,2k+1) * 2^k = Sum_{k=0..n} A034867(n,k) * 2^k = (1/n!) * Sum_{k=0..n} A131980(n,k) * 2^k. - _Tom Copeland_, Nov 30 2007

%F Equals row sums of unsigned triangle A133156. - _Gary W. Adamson_, Apr 21 2008

%F a(n) (n >= 3) is the determinant of the (n-1) X (n-1) tridiagonal matrix with diagonal entries 2, superdiagonal entries 1 and subdiagonal entries -1. - _Emeric Deutsch_, Aug 29 2008

%F a(n) = 5*a(n-2) + 2*a(n-3), a(n) = 6*a(n-2) - a(n-4). - _Mohamed Bouhamida_, Sep 04 2008

%F a(n) = A000045(n) + Sum_{k=1..n-1} A000045(k)*a(n-k). - _Roger L. Bagula_ and _Gary W. Adamson_, Sep 07 2008

%F From _Hieronymus Fischer_, Jan 02 2009: (Start)

%F fract((1+sqrt(2))^n)) = (1/2)*(1 + (-1)^n) - (-1)^n*(1+sqrt(2))^(-n) = (1/2)*(1 + (-1)^n) - (1-sqrt(2))^n.

%F See A001622 for a general formula concerning the fractional parts of powers of numbers x > 1, which satisfy x - x^(-1) = floor(x).

%F a(n) = round((1+sqrt(2))^n) for n > 0. (End)

%F a(n)=((4+sqrt(18))*(1+sqrt(2))^n) + (4-sqrt(18))*(1-sqrt(2))^n)/4 offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

%F If p[i] = Fibonacci(i) and if A is the Hessenberg matrix of order n defined by A[i,j] = p[j-i+1] when i<=j, A[i,j]=-1 when i=j+1, and A[i,j]=0 otherwise, then, for n >= 1, a(n) = det A. - _Milan Janjic_, May 08 2010

%F a(n) = 3*a(n-1) - a(n-2) - a(n-3), n > 2. - _Gary Detlefs_, Sep 09 2010

%F a(n) = 2*(a(2k-1) + a(2k))*a(n-2k) - a(n-4k).

%F a(n) = 2*(a(2k) + a(2k+1))*a(n-2k-1) + a(n-4k-2). - _Charlie Marion_, Apr 13 2011

%F G.f.: x/(1 - 2*x - x^2) = sqrt(2)*G(0)/4; G(k) = ((-1)^k) - 1/(((sqrt(2) + 1)^(2*k)) - x*((sqrt(2) + 1)^(2*k))/(x + ((sqrt(2) - 1)^(2*k + 1))/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Dec 02 2011

%F In general, for n > k, a(n) = a(k+1)*a(n-k) + a(k)*a(n-k-1). See definition of Pell numbers and the formula for Sep 04 2008. - _Charlie Marion_, Jan 17 2012

%F a(n) = A216134(2*floor(n/2) + 1) - (2 - n(mod 2))*A216134(2*floor(n/2)) + (1 - n(mod 2))*A216134(2*floor(n/2) - 1); A216134 gives the indices of the Sophie Germain triangular numbers. - _Raphie Frank_, Jan 04 2013

%F Sum{n>=1} (-1)^(n-1)/(a(n)*a(n+1)) = sqrt(2) - 1. - _Vladimir Shevelev_, Feb 22 2013

%F From _Vladimir Shevelev_, Feb 24 2013: (Start)

%F (1) Expression a(n+1) via a(n): a(n+1) = a(n) + sqrt(2*a^2(n) + (-1)^n);

%F (2) a(n+1)^2 - a(n)*a(n+2) = (-1)^n;

%F (3) Sum_{k=1..n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);

%F (4) a(n)/a(n+1) = sqrt(2) - 1 + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (End)

%F a(-n) = -(-1)^n * a(n). - _Michael Somos_, Jun 01 2013

%F G.f.: G(0)/(2+2*x) - 1/(1+x), where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Aug 10 2013

%F G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 30 2013

%F a(n) = Sum_{r=0..n-1} Sum_{k=0..n-r-1} binomial(r+k,k)*binomial(k,n-k-r-1). - _Peter Luschny_, Nov 16 2013

%F a(n) = Sum_{k=1,3,5,...<=n} C(n,k)*2^((k-1)/2). - _Vladimir Shevelev_, Feb 06 2014

%F a(2n) = 2*a(n)*(a(n-1) + a(n)). - _John Blythe Dobson_, Mar 08 2014

%F a(k*n) = a(k)*a(k*n-k+1) + a(k-1)*a(k*n-k). - _Charlie Marion_, Mar 27 2014

%F a(k*n) = 2*a(k)*(a(k*n-k)+a(k*n-k-1)) + (-1)^k*a(k*n-2k). - _Charlie Marion_, Mar 30 2014

%F a(n+1) = (1+sqrt(2))*a(n) + (1-sqrt(2))^n. - _Art DuPre_, Apr 04 2014

%F a(n+1) = (1-sqrt(2))*a(n) + (1+sqrt(2))^n. - _Art DuPre_, Apr 04 2014

%F a(n) = F(n) + Sum_{k=1..n} F(k)*a(n-k), n >= 0 where F(n) the Fibonacci numbers A000045. - _Ralf Stephan_, May 23 2014

%F a(n) = round(sqrt(a(2n) + a(2n-1)))/2. - _Richard R. Forberg_, Jun 22 2014

%F a(n) = Product_{k divides n} A008555(k). - _Tom Edgar_, Jan 28 2015

%F a(n+k)^2 - A002203(k)*a(n)*a(n+k) + (-1)^k*a(n)^2 = (-1)^n*a(k)^2. - _Alexander Samokrutov_, Aug 06 2015

%F a(n) = 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1)) for n >= 2. - _Peter Luschny_, Dec 17 2015

%F a(n+1) = Sum_{k=0..n} binomial(n,k)*2^floor(k/2). - _Tony Foster III_, May 07 2017

%F a(n) = exp((i*Pi*n)/2)*sinh(n*arccosh(-i))/sqrt(2). - _Peter Luschny_, Mar 07 2018

%F From _Rogério Serôdio_, Mar 30 2018: (Start)

%F Some properties:

%F (1) a(n)^2 - a(n-2)^2 = 2*a(n-1)*(a(n) + a(n-2)) (see A005319);

%F (2) a(n-k)*a(n+k) = a(n)^2 + (-1)^(n+k+1)*a(k)^2;

%F (3) Sum_{k=2..n+1} a(k)*a(k-1) = a(n+1)^2 if n is odd, else a(n+1)^2 - 1 if n is even;

%F (4) a(n) - a(n-2*k+1) = (A077444(k) - 1)*a(n-2*k+1) + a(n-4*k+2);

%F (5) Sum_{k=n..n+9} a(k) = 41*A001333(n+5). (End)

%F From _Kai Wang_, Dec 30 2019: (Start)

%F a(m+r)*a(n+s) - a(m+s)*a(n+r) = -(-1)^(n+s)*a(m-n)*a(r-s).

%F a(m+r)*a(n+s) + a(m+s)*a(n+r) = (2*A002203(m+n+r+s) - (-1)^(n+s)*A002203(m-n)*A002203(r-s))/8.

%F A002203(m+r)*A002203(n+s) - A002203(m+s)*A002203(n+r) = (-1)^(n+s)*8*a(m-n)*a(r-s).

%F A002203(m+r)*A002203(n+s) - 8*a(m+s)*a(n+r) = (-1)^(n+s)*A002203(m-n)*A002203(r-s).

%F A002203(m+r)*A002203(n+s) + 8*a(m+s)*a(n+r) = 2*A002203(m+n+r+s)+ (-1)^(n+s)*8*a(m-n)*a(r-s). (End)

%F From _Kai Wang_, Jan 12 2020: (Start)

%F a(n)^2 - a(n+1)*a(n-1) = (-1)^(n-1).

%F a(n)^2 - a(n+r)*a(n-r) = (-1)^(n-r)*a(r)^2.

%F a(m)*a(n+1) - a(m+1)*a(n) = (-1)^n*a(m-n).

%F a(m-n) = (-1)^n (a(m)*A002203(n) - A002203(m)*a(n))/2.

%F a(m+n) = (a(m)*A002203(n) + A002203(m)*a(n))/2.

%F A002203(n)^2 - A002203(n+r)*A002203(n-r) = (-1)^(n-r-1)*8*a(r)^2.

%F A002203(m)*A002203(n+1) - A002203(m+1)*A002203(n) = (-1)^(n-1)*8*a(m-n).

%F A002203(m-n) = (-1)^(n)*(A002203(m)*A002203(n) - 8*a(m)*a(n) )/2.

%F A002203(m+n) = (A002203(m)*A002203(n) + 8*a(m)*a(n) )/2. (End)

%e G.f. = x + 2*x^2 + 5*x^3 + 12*x^4 + 29*x^5 + 70*x^6 + 169*x^7 + 408*x^8 + 985*x^9 + ...

%p A000129 := proc(n) option remember; if n <=1 then n; else 2*procname(n-1)+procname(n-2); fi; end;

%p a := n-> (Matrix([[2,1],[1,0]])^n)[1,2]: seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 01 2008

%p A000129 := n -> `if`(n<2, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1)):

%p seq(simplify(A000129(n)), n=0..31); # _Peter Luschny_, Dec 17 2015

%t CoefficientList[Series[x/(1 - 2*x - x^2), {x, 0, 60}], x] (* _Stefan Steinerberger_, Apr 08 2006 *)

%t Expand[Table[((1 + Sqrt[2])^n - (1 - Sqrt[2])^n)/(2Sqrt[2]), {n, 0, 30}]] (* _Artur Jasinski_, Dec 10 2006 *)

%t LinearRecurrence[{2, 1}, {0, 1}, 60] (* _Harvey P. Dale_, Jan 04 2012 *)

%t a[ n_] := With[ {s = Sqrt@2}, ((1 + s)^n - (1 - s)^n) / (2 s)] // Simplify; (* _Michael Somos_, Jun 01 2013 *)

%t Table[Fibonacci[n, 2], {n, 0, 20}] (* _Vladimir Reshetnikov_, May 08 2016 *)

%t Fibonacci[Range[0, 20], 2] (* _Eric W. Weisstein_, Sep 30 2017 *)

%o (PARI) default(realprecision, 2000); for (n=0, 4000, a=contfracpnqn(vector(n, i, 1+(i>1)))[2, 1]; if (a > 10^(10^3 - 6), break); write("b000129.txt", n, " ", a)); \\ _Harry J. Smith_, Jun 12 2009

%o (PARI) {a(n) = imag( (1 + quadgen( 8))^n )}; /* _Michael Somos_, Jun 01 2013 */

%o (PARI) {a(n) = if( n<0, -(-1)^n, 1) * contfracpnqn( vector( abs(n), i, 1 + (i>1))) [2, 1]}; /* _Michael Somos_, Jun 01 2013 */

%o (PARI) a(n)=([2, 1; 1, 0]^n)[2,1] \\ _Charles R Greathouse IV_, Mar 04 2014

%o (Sage) [lucas_number1(n, 2, -1) for n in range(30)] # _Zerinvary Lajos_, Apr 22 2009 (Haskell)

%o a000129 n = a000129_list !! n

%o a000129_list = 0 : 1 : zipWith (+) a000129_list (map (2 *) $ tail a000129_list)

%o -- _Reinhard Zumkeller_, Jan 05 2012, Feb 05 2011

%o (Maxima)

%o a[0]:0$

%o a[1]:1$

%o a[n]:=2*a[n-1]+a[n-2]$

%o A000129(n):=a[n]$

%o makelist(A000129(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */

%o (Maxima) makelist((%i)^(n-1)*ultraspherical(n-1,1,-%i),n,0,24),expand; /* _Emanuele Munarini_, Mar 07 2018 */

%o (MAGMA) [0] cat [n le 2 select n else 2*Self(n-1) + Self(n-2): n in [1..35]]; // _Vincenzo Librandi_, Aug 08 2015

%o (GAP) a := [0,1];; for n in [3..10^3] do a[n] := 2 * a[n-1] + a[n-2]; od; A000129 := a; # _Muniru A Asiru_, Oct 16 2017

%Y Partial sums of A001333.

%Y 2nd row of A172236.

%Y a(n) = A054456(n-1, 0), n>=1 (first column of triangle).

%Y Cf. A002203, A096669, A096670, A097075, A097076, A051927, A005409.

%Y Cf. A175181 (Pisano periods), A214028 (Entry points), A214027 (number of zeros in a fundamental period).

%Y A077985 is a signed version.

%Y INVERT transform of Fibonacci numbers (A000045).

%Y Cf. A038207.

%Y The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

%Y Cf. A034867, A131980, A133156, A143808, A135387, A153346, A001622, A006497, A014176, A098316, A154325, A021083, A216134, A243399, A008555.

%Y Cf. A048739.

%Y Cf. A073133

%K nonn,easy,core,cofr,nice,frac,changed

%O 0,3

%A _N. J. A. Sloane_

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Last modified February 16 22:46 EST 2020. Contains 331976 sequences. (Running on oeis4.)