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A000129
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Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
(Formerly M1413 N0552)
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456
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0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962, 7645370045, 18457556052, 44560482149, 107578520350, 259717522849
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OFFSET
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0,3
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COMMENTS
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Sometimes also called lambda numbers.
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
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
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.
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
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 (amarnath_murthy(AT)yahoo.com), Mar 22 2003
This is also the Horadam sequence (0,1,1,2). a(n) / a(n-1) converges to 2^1/2 + 1 as n approaches infinity. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
Number of 132-avoiding two-stack sortable permutations.
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
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
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
Apart from initial terms, Pisot sequence P(2,5). See A008776 for definition of Pisot sequences. - David W. Wilson
Sums of antidiagonals of A038207 [Pascal's triangle squared] - Ross La Haye (rlahaye(AT)new.rr.com), Oct 28 2004
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
a(n) = sum of n-th row of triangle in A008288 = A094706(n)+A000079(n). - Reinhard Zumkeller, Dec 03 2004
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 1 2006
(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
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*(.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) = .41421356... = [2, 2, 2,...], the convergents being [1/2, 2/5, 5/12,...]. - Gary W. Adamson, Dec 21 2007
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
Prime Pell numbers with an odd index gives the RMS value (A141812) of prime RMS numbers (A140480). [From Ctibor O. Zizka, Aug 13 2008]
Comment from Clark Kimberling, Aug 27 2008 (Start): Related convergents (numerator/denominator):
lower principal convergents: A002315/A001653
upper principal convergents: A001541/A001542
intermediate convergents: A052542/A001333
lower intermediate convergents: A005319/A001541
upper intermediate convergents: A075870/A002315
principal and intermediate convergents: A143607/A002965
lower principal and intermediate convergents: A143608/A079496
upper principal and intermediate convergents: A143609/A084068 (End)
Equals row sums of triangle A143808 starting with offset 1. [From Gary W. Adamson, Sep 01 2008]
Binomial transform of the sequence:= 0,1,0,2,0,4,0,8,0,16,..., powers of 2 alternating with zeros. [From Philippe DELEHAM, Oct 28 2008]
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. [From Patrick Costello (pat.costello(AT)eku.edu), Dec 07 2008]
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. [From Gary W. Adamson, Dec 29 2008]
Starting with offset 1 = row sums of triangle A153345 [From Gary W. Adamson, Dec 24 2008]
Contribution from Charlie Marion, Jan 07 2009: (Start)
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:
let a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2)
and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);
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)
and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).
For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5, 17/12, 41/29.
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
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
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);
for example, if k=1 and n=3, then a(1,n)=a(n+1) and
1*a(1,6)^2-a(1,5)*a(1,7)=1*169^2-70*408=1;
1*a(1,4)*a(1,6)-a(1,5)^2=1*29*169-70^2=1;
b(1,5)*b(1,7)-1*b(1,6)^2=99*577-1*239^2=2;
b(1,5)^2-1*b(1,4)*b(1,6)=99^2-1*41*239=2.
Cf. A001333, A142238-A142239, A153313-153318.
[From Charlie Marion, Jan 07 2009]
(End)
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,...). [From Gary W. Adamson, Jan 18 2009]
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)). [From Gary W. Adamson, Feb 21 2009]
Equals eigensequence of triangle A154325 [From Gary W. Adamson, Feb 12 2009]
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 by n rectangle with indistinguishable 1 by 2 rectangles and 1 by 1 squares that come in two varieties, A and B say. For example, with C representing the 1 by 2 rectangle, we obtain a(4)=12 from AAA, AAB, ABA, BAA, ABB, BAB, BBA, BBB, AC, BC, CA and CB. [From Martin Griffiths (griffm(AT)essex.ac.uk), Apr 25 2009]
a(n+1)=2*a(n)+ a(n-1) a(1=1,a(2)=2 was used by Theon from Smyrna. [From Sture Sjoestedt (sture.sjostedt(AT)spray.se), May 29 2009]
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. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
Number of units of a(n) belongs to a periodic sequence: 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
As a fraction: 1/79 = 0.0126582278481... or 1/9799 = 0.000102051229...(1/119 and 1/10199 for sequence in reverse). [From M. Dols (markdols99(AT)yahoo.com), May 18 2010]
Contribution from Gary W. Adamson, Jul 16 2010: (Start)
As n->inf. (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... (End)
Numbers n such that 2*n^2+-1 is a square. [From Vincenzo Librandi, Jul 18 2010]
Starting (1, 2, 5,...) = INVERTi transform of A006190: (1, 3, 10, 33, 109,...). [From Gary W. Adamson, Aug 06 2010]
[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. [From James R. Buddenhagen, Aug 14 2010]
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. [From Johannes W. Meijer, Aug 15 2010]
Let the 2x2 square matrix A=[2, 1; 1, 0] then a(n) = the (1,1) element of A^(n-1). [Carmine Suriano, Jan 14 2011]
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)*2^.5 and each of the coordinates of its point of intersection with C(n + 1) is a(2n + 1) + (A001333(2n + 1)*2^.5)/2. See similar Comments for A001109 and A001653, Sep 14 2005. - Charlie Marion, Jan 18 2012
A001333 and A000129 give the diagonal numbers described by Theon from Smyrna - Sture Sjöstedt, Oct 20 2012
Pell numbers could be named also "silver Fibonacci numbers", since, for n>=1, F(n+1)=ceil(phi*F(n)), if n is even and F(n+1)=floor(phi*F(n)), if n is odd, where phi is golden ratio, while a(n+1)=ceil(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 silver ratio. - Vladimir Shevelev, Feb 22 2013
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REFERENCES
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Ayoub B. Ayoub, "Fibonacci-like sequences and Pell equations", The College Mathematics Journal, Vol. 38 (2007), pp. 49-53.
P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.
M. Barnabei, F. Bonetti and M. Silimbani, Two permutation classes related to the Bubble Sort operator, Electronic Journal of Combinatorics 19(3) (2012), #P25. -
From N. J. A. Sloane, Dec 25 2012
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
S.-M. Belcastro, Tilings of 2 x n Grids on Surfaces, preprint. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
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.
John Derbyshire, Prime Obsession, Joseph Henry Press, 2004, see p. 16.
E. Deutsch, A formula for the Pell numbers, Problem 10663, Amer. Math. Monthly 107 (No. 4, 2000), solutions pp. 370-371.
E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Ex.1, p. 237-8.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.1.
R. P. Grimaldi, Ternary strings ..., Congressus Numerantium, 205 (2011), 129-149.
A. F. Horadam, Special Properties of the Sequence W(n){a, b; p, q}, Fibonacci Quarterly, Vol. 5, No. 5, 1967, pp. 424-434.
A. F. Horadam, Pell identities, Fib. Quart., 9 (1971), 245-252, 263.
Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.
S. Kitaev, J. Remmel and M. Tiefenbruck, Quadrant marked mesh patterns in 132-avoiding permutations II, arXiv preprint arXiv:1302.2274, 2013
Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
Hao Pan, Arithmetic properties of q-Fibonacci numbers and q-Pell numbers, Discr. Math., 306 (2006), 2118-2127. [From N. J. A. Sloane, Jan 29 2009]
Problem B-82, Fib. Quart., 4 (1966), 374-375.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 43.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
Manfred R. Schroeder, "Number Theory in Science and Communication", 5-th ed., Springer-Verlag, 2009, p. 90. [From Gary W. Adamson, Feb 21 2009]
Mark A. Shattuck, Tiling proofs of some formulas for the Pell numbers of odd index, Integers, 9 (2009), 53-64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see Eq. (20)).
Author?, Extending Theon's Ladder to Any Square Root, Problem 3858, Elementa, No. 4 1996 [From Sture Sjoestedt (sture.sjostedt(AT)spray.se), May 29 2009]
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..500
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
C. Raissi and J. Pei, Towards Bounding Sequential Patterns.
Ian Walker, Explorations in Recursion with John Pell and the Pell Sequence.
Tewodros Amdeberhan, Solution to problem #10663 (AMM)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. S. Egge and T. Mansour, 132-avoiding two-stack sortable permutations....
Nick Hobson, Python program for this sequence
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 135
Tanya Khovanova, Recursive Sequences
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368, 2012
James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2
R. A. Sulanke, Moments, Narayana numbers and the cut and paste for lattice paths
Eric Weisstein's World of Mathematics, Pell Number
Eric Weisstein's World of Mathematics, Pell Polynomial
Eric Weisstein's World of Mathematics, Square Root
Eric Weisstein's World of Mathematics, Pythagoras's Constant
Eric Weisstein's World of Mathematics, Square Triangular Number
Index entries for "core" sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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G.f.: x/(1-2*x-x^2).
a(n) = 2*a(n-1)+a(n-2), a(0)=0, a(1)=1.
a(n)=( (1+sqrt(2))^n -(1-sqrt(2))^n )/(2*sqrt(2))
a(n) = integer nearest a(n-1)/(sqrt(2) - 1), where a(0) = 1 - from Clark Kimberling
a(n)= Sum_{i, j, k >= 0: i+j+2k=n} (i+j+k)!/(i!*j!*k!).
a(n)^2 + a(n+1)^2 = a(2n+1) (1999 Putnam examination).
a(2n) = 2*a(n)*A001333(n). - John McNamara, Oct 30, 2002
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.
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
a(n)=sum{k=0, ..floor(n/2), C(n, 2k+1)2^k}. - Paul Barry, May 13 2003
a(n-2) + a(n) = (1 + sqrt2)^(n-1) + (1 - sqrt2)^(n-1) = A002203(n-1). [A002203(n)]^2 - 8[a(n)]^2 = 4(-1)^n - Gary W. Adamson, Jun 15 2003
Nonreduced 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
a(n+1)=Sum(C(n-k, k)2^(n-2k), k=0, .., Floor[n/2]). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
Apart from initial terms, inverse binomial transform of A052955. - Paul Barry, May 23 2004
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
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
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
a(n)=F(n, 2), the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006
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 (creighton.k.dement(AT)uni-oldenburg.de), Jul 21 2005
a(r+s) = a(r)*a(s+1) + a(r-1)*a(s). - Lekraj Beedassy, Sep 03 2006
a(n)=(b(n+1)+b(n-1))/n where {b(n)} is the sequence A006645 - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Nov 22 2006
Comments from Miklos Kristof, Mar 19 2007: (Start)
Let F(n)=a(n)=Pell numbers, L(n)=A002203=companion Pell numbers (A002203):
For a>=b and odd b F(a+b)+F(a-b)=L(a)*F(b).
For a>=b and even b F(a+b)+F(a-b)=F(a)*L(b).
For a>=b and odd b F(a+b)-F(a-b)=F(a)*L(b).
For a>=b and even b F(a+b)-F(a-b)=L(a)*F(b).
F(n+m)+(-1)^m*F(n-m)=F(n)*L(m)
F(n+m)-(-1)^m*F(n-m)=L(n)*F(m)
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(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(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(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)
a(n+1)*a(n)=2*sum{k=0..n, a(k)^2} (a similar relation holds for A001333) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 28 2007
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
Equals row sums of unsigned triangle A133156. - Gary W. Adamson, Apr 21 2008
a(n) (n>=3) is the determinant of the (n-1) by (n-1) tridiagonal matrix with diagonal entries 2, superdiagonal entries 1 and subdiagonal entries -1. [From Emeric Deutsch, Aug 29 2008]
a(n) = 5*a(n-2)+2*a(n-3), a(n) = 6*a(n-2)-a(n-4). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2008
a(n) = A000045(n) + sum_{k=1..n-1} A000045(k)*a(n-k). - Roger L. Bagula and Gary W. Adamson, Sep 07 2008
Comments from Hieronymus Fischer, Jan 02 2009 (Start): 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.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which suffice x-x^(-1)=floor(x).
a(n) = round((1+sqrt(2))^n) for n>0. (End)
a(n)=((4+sqrt18)*(1+sqrt2)^n)+(4-sqrt18)*(1-sqrt2)^n)/4 offset 0. [From Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009]
If p[i]=fibonacci(i) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. [From Milan Janjic, May 08 2010]
a(n)= 3*a(n-1)-a(n-2)-a(n-3),n>2.- Gary Detlefs, Sep 09 2010
a(n)=2*(a(2k-1)+a(2k))*a(n-2k)-a(n-4k).
a(n)=2*(a(2k)+a(2k+1))*a(n-2k-1)+a(n-4k-2). - Charlie Marion, Apr 13 2011
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
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
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
sum{n>=1}(-1)^(n-1)/(a(n)*a(n+1))=sqrt(2)-1. - Vladimir Shevelev, Feb 22 2013
From Vladimir Shevelev, Feb 24 2013: (Start)
(1) Expression a(n+1) via a(n): a(n+1) = a(n) + sqrt(2*a^2(n) + (-1)^n);
(2) a^2(n+1) - a(n)*a(n+2) = (-1)^n;
(3) sum_{k=1,...,n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);
(4) a(n)/a(n+1) = sqrt(2)-1 + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (end)
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MAPLE
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A000129 := proc(n) option remember; if n <=1 then n; else 2*A000129(n-1)+A000129(n-2); fi; end;
with(numtheory):pel := cfrac (sin(Pi/4), 100): seq(nthnumer(pel, i), i=0..29 ); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2007
A000129:=-1/(-1+2*z+z**2); [Simon Plouffe in his 1992 dissertation.]
a:= n-> (Matrix([[2, 1], [1, 0]])^n)[1, 2]: seq (a(n), n=0..40); # Alois P. Heinz, Aug 01 2008
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MATHEMATICA
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CoefficientList[Series[x/(1 - 2*x - x^2), {x, 0, 60}], x] - Stefan Steinerberger, Apr 08 2006
Expand[Table[((1 + Sqrt[2])^n - (1 - Sqrt[2])^n)/(2Sqrt[2]), {n, 0, 30}]] - Artur Jasinski, Dec 10 2006
a=1; b=0; c=0; lst={b}; Do[c=a+b+c; AppendTo[lst, c]; a=b; b=c, {n, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Mar 23 2009]
LinearRecurrence[{2, 1}, {0, 1}, 60] (* From Harvey P. Dale, Jan 04 2012 *)
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PROG
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(PARI) a(n)=if(n<0, 0, contfracpnqn(vector(n, i, 1+(i>1)))[2, 1])
(Sage) [lucas_number1(n, 2, -1) for n in xrange(0, 30)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(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); ); } [From Harry J. Smith, Jun 12 2009]
(Haskell)
a000129 n = a000129_list !! n
a000129_list = 0 : 1 : zipWith (+) a000129_list (map (2 *) $ tail a000129_list)
-- Reinhard Zumkeller, 05 Jan 2012, Feb 05 2011
(Maxima)
a[0]:0$
a[1]:1$
a[n]:=2*a[n-1]+a[n-2]$
A000129(n):=a[n]$
makelist(A000129(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */
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CROSSREFS
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Partial sums of A001333.
a(n) = A054456(n-1, 0), n>=1 (first column of triangle).
Cf. A002203, A096669, A096670, A097075, A097076, A051927, A005409, A175181 (Pisano periods).
A077985 is a signed version.
INVERT transform of Fibonacci numbers (A000045).
Cf. A038207.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A034867, A131980.
Cf. A133156.
Cf. A143808 [From Gary W. Adamson, Sep 01 2008]
Cf. A135387, A153346. [From Gary W. Adamson, Dec 29 2008]
Cf. A001622, A006497, A014176, A098316.
Cf. A154325 [From Gary W. Adamson, Feb 12 2009]
Cf. A021083 [From M. Dols (markdols99(AT)yahoo.com), May 18 2010]
Cf. A216134 [From Raphie Frank , Jan 04 2013]
Sequence in context: A215936 * A077985 A215928 A054198 A054196 A131710
Adjacent sequences: A000126 A000127 A000128 * A000130 A000131 A000132
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KEYWORD
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nonn,easy,core,cofr,nice,frac
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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