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 A001333 Numerators of continued fraction convergents to sqrt(2). (Formerly M2665 N1064) 315

%I M2665 N1064

%S 1,1,3,7,17,41,99,239,577,1393,3363,8119,19601,47321,114243,275807,

%T 665857,1607521,3880899,9369319,22619537,54608393,131836323,318281039,

%U 768398401,1855077841,4478554083,10812186007,26102926097,63018038201,152139002499

%N Numerators of continued fraction convergents to sqrt(2).

%C Number of n-step non-selfintersecting paths starting at (0,0) with steps of types (1,0), (-1,0) or (0,1) [Stanley].

%C Number of n steps one-sided prudent walks with east, west and north steps. - _Shanzhen Gao_, Apr 26 2011

%C Number of ternary strings of length n-1 with subwords (0,2) and (2,0) not allowed. - _Olivier Gérard_, Aug 28 2012

%C Number of symmetric 2n X 2 or (2n-1) X 2 crossword puzzle grids: all white squares are edge connected; at least 1 white square on every edge of grid; 180 degree rotational symmetry. - _Erich Friedman_

%C a(n+1) is the number of ways to put molecules on a 2 X n ladder lattice so that the molecules do not touch each other.

%C In other words, a(n+1) is the number of independent vertex sets and vertex covers in the n-ladder graph P_2 X P_n. - _Eric W. Weisstein_, Apr 04 2017

%C Number of (n-1) X 2 binary arrays with a path of adjacent 1's from top row to bottom row. - _R. H. Hardin_, Mar 16 2002

%C a(2*n+1) with b(2*n+1) := A000129(2*n+1), n>=0, give all (positive integer) solutions to Pell equation a^2 - 2*b^2 = -1.

%C a(2*n) with b(2*n) := A000129(2*n), n>=1, give all (positive integer) solutions to Pell equation a^2 - 2*b^2 = +1 (see Emerson reference).

%C Bisection: a(2*n) = T(n,3) = A001541(n), n>=0 and a(2*n+1) = S(2*n,2*sqrt(2)) = A002315(n), n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310.

%C Binomial transform of A077957. - _Paul Barry_, Feb 25 2003

%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) = 2. - _Herbert Kociemba_, Jun 02 2004

%C For n>1, a(n) corresponds to the longer side of a near right-angled isosceles triangle, one of the equal sides being A000129(n). - _Lekraj Beedassy_, Aug 06 2004

%C Exponents of terms in the series F(x,1), where F is determined by the equation F(x,y) = xy + F(x^2*y,x). - _Jonathan Sondow_, Dec 18 2004

%C Number of n-words from the alphabet A={0,1,2} which two neighbors differ by at most 1. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 30 2006

%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 numerators. - _Amarnath Murthy_, Mar 22 2003 [Amended by Paul E. Black (paul.black(AT)nist.gov), Dec 18 2006]

%C a(n) mod 10 = A131707. See A131711. - _Paul Curtz_, Apr 08 2008

%C Prime numerators with an odd index are prime RMS numbers(A140480) and also NSW primes(A088165). - _Ctibor O. Zizka_, Aug 13 2008

%C The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17, 58/41; essentially, numerators=A052542 and denominators here. - _Clark Kimberling_, Aug 26 2008

%C Equals right border of triangle A143966. Starting (1, 3, 7, ...) equals INVERT transform of (1, 2, 2, 2, ...) and row sums of triangle A143966. - _Gary W. Adamson_, Sep 06 2008

%C Inverse binomial transform of A006012; Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, ...]. - _Philippe Deléham_, Dec 04 2008

%C Contribution 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) 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) 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 b(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. A000129, A142238-A142239, A153313-A153318.

%C (End)

%C This sequence occurs in the lower bound of the order of the set of equivalent resistances of n equal resistors combined in series and in parallel (A048211). - _Sameen Ahmed Khan_, Jun 28 2010

%C Let M = a triangle with the Fibonacci series in each column, but the leftmost column is shifted upwards one row. A001333 = lim_{n->inf} M^n, the left-shifted vector considered as a sequence. - _Gary W. Adamson_, Jul 27 2010

%C a(n) is the number of compositions of n when there are 1 type of 1 and 2 types of other natural numbers. - _Milan Janjic_, Aug 13 2010

%C Equals the INVERTi transform of A055099. - _Gary W. Adamson_, Aug 14 2010

%C From _L. Edson Jeffery_, Apr 04 2011: (Start)

%C Let U be the unit-primitive matrix (see [Jeffery])

%C U=U_(8,2)=

%C (0 0 1 0)

%C (0 1 0 1)

%C (1 0 2 0)

%C (0 2 0 1).

%C (End)

%C For n>=1, row sums of triangle

%C m/k.|..0.....1.....2.....3.....4.....5.....6.....7

%C ==================================================

%C .0..|..1

%C .1..|..1.....2

%C .2..|..1.....2.....4

%C .3..|..1.....4.....4.....8

%C .4..|..1.....4....12.....8....16

%C .5..|..1.....6....12....32....16....32

%C .6..|..1.....6....24....32....80....32....64

%C .7..|..1.....8....24....80....80...192....64...128

%C which is triangle for numbers 2^k*C(m,k) with duplicated diagonals. - _Vladimir Shevelev_, Apr 12 2012

%C a(n) is also the number of ways to place k non-attacking wazirs on a 2 X n board, summed over all k>=0 (a wazir is a leaper [0,1]). - _Vaclav Kotesovec_, May 08 2012

%C The sequences a(n) and b(n) := A000129(n) are entries of

%C powers of the special case of the Brahmagupta Matrix - for details see Suryanarayan's paper. Further, as Suryanarayan remark, if we set A = 2*(a(n) + b(n))*b(n), B = a(n)*(a(n) + 2*b(n)), C = a(n)^2 + 2*a(n)*b(n) + 2*b(n)^2 we obtain integral solutions of the Pythagorean relation A^2 + B^2 = C^2, where A and B are consecutive integers. - _Roman Witula_, Jul 28 2012

%C Pisano period lengths: 1, 1, 8, 4, 12, 8, 6, 4, 24, 12, 24, 8, 28, 6, 24, 8, 16, 24, 40, 12, .... - _R. J. Mathar_, Aug 10 2012

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

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

%C If p is prime, a(p) == 1 mod p (compare with similar comment for A000032). - _Creighton Dement_, Oct 11 2005, modified by _Davide Colazingari_, Jun 26 2016

%D Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

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

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

%D Reinhardt Euler, The Fibonacci Number of a Grid Graph and a New Class of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.6.

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

%D S. Gao, H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).

%D David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.

%D Martin Griffiths, Pell identities via a quadratic field, International Journal of Mathematical Education in Science and Technology, 2013, DOI:10.1080/0020739X.2013.790512.

%D R. P. Grimaldi, Ternary strings ..., Congressus Numerantium, 205 (2011), 129-149.

%D Hoots, Lucas, "Strong quota pair systems and May's theorem on median semilattices." (2015). Univ. Louisville, Electronic Theses and Dissertations. Paper 2253. http://dx.doi.org/10.18297/etd/2253

%D A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.

%D Y. Kong, Ligand binding on ladder lattices, Biophysical Chemistry, Vol. 81 (1999), pp. 7-21.

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

%D Kuba, Markus; Panholzer, Alois. Enumeration formulas for pattern restricted Stirling permutations. Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012

%D Kin Y. Li, Math Problem Book I, 2001, p. 24, Problem 159.

%D Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.

%D I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 102, Problem 10.

%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.

%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).

%D R. P. Stanley, Enumerative Combinatorics, Volume 1 (1986), p. 203, Example 4.1.2.

%D A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

%D 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. (21)).

%D Gy. Tasi et al., Quantum algebraic-combinatoric study of the conformational properties of n-alkanes. II, J. Math. Chemistry, 27, 2000, 191-199 (see p. 193).

%D V. Thebault, Concerning two classes of remarkable perfect square pairs, Amer. Math. Monthly, 56 (1949), 443-448.

%D R. C. Tilley et al., The cell growth problem for filaments, Proc. Louisiana Conf. Combinatorics, ed. R. C. Mullin et al., Baton Rouge, 1970, 310-339.

%H T. D. Noe, <a href="/A001333/b001333.txt">Table of n, a(n) for n = 0..500</a>

%H Antoni Amengual, <a href="http://dx.doi.org/10.1119/1.19396">The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel</a>, American Journal of Physics, 68(2), (2000) 175-179. [From _Sameen Ahmed Khan_, Jun 28 2010]

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp.313-315

%H C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.

%H S. Barbero, U. Cerruti, N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero2/barbero7.html">A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences </a>, J. Int. Seq. 13 (2010) # 10.9.7

%H Florin P. Boca, Christopher Linden, <a href="https://arxiv.org/abs/1705.01238">On Minkowski type question mark functions associated with even or odd continued fractions</a>, arXiv:1705.01238 [math.DS], 2017. See p. 7.

%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 Fan Chung, R. L. Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Am. Math. Monthly 115 (3) (2008) 185-194

%H K. Dohmen, <a href="http://arxiv.org/abs/1403.0969">Closed-form expansions for the universal edge elimination polynomial</a>, arXiv preprint arXiv:1403.0969 [math.CO], 2014.

%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, see Ex. 1, pp. 237-238.

%H Shanzhen Gao, Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2696.pdf">Tackling Sequences From Prudent Self-Avoiding Walks</a>, FCS'14, The 2014 International Conference on Foundations of Computer Science.

%H F. Harary and R. W. Robinson, <a href="/A001333/a001333_2.pdf">Tapeworms</a>, Unpublished manuscript, circa 1973. (Annotated scanned copy)

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

%H Nick Hobson, <a href="/A001333/a001333.py.txt">Python program for this sequence</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=143">Encyclopedia of Combinatorial Structures 143</a>

%H A. F. Horadam, R. P. Loh and A. G. Shannon, <a href="/A005013/a005013.pdf">Divisibility properties of some Fibonacci-type sequences</a>, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979. [Annotated scanned copy]

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

%H L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices)</a>

%H Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1004.3346">The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel</a>, arXiv:1004.3346 [From _Sameen Ahmed Khan_, Jun 28 2010]

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

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 392-3.

%H Dmitry Kruchinin, <a href="http://arxiv.org/abs/1211.2100">Integer properties of a composition of exponential generating functions</a>, arXiv:1211.2100

%H J. V. Leyendekkers and A. G. Shannon, <a href="http://nntdm.net/volume-18-2012/number-2/58-62/">Pellian sequence relationships among pi, e, sqrt(2)</a>, Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62. See Table 2. - _N. J. A. Sloane_, Dec 23 2012

%H Kin Y. Li, <a href="http://www.math.ust.hk/faculty/makyli/190_2006Su/problembk.ps">Problem 159</a>

%H Barry Mazur, <a href="https://doi.org/10.1090/S0273-0979-1986-15430-3">Arithmetic on curves</a>, Bull. Amer. Math. Soc. 14 (1986), 207-259.

%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 H. Prodinger and R. F. Tichy, <a href="http://www.fq.math.ca/Scanned/20-1/prodinger.pdf">Fibonacci numbers of graphs</a>, Fibonacci Quarterly, 20, 1982, 16-21.

%H Jon E. Schoenfield, <a href="/A001333/a001333.txt">Prime factorization of a(n) for n = 0..300</a>

%H B. Srinivasa Rao, <a href="http://www.fq.math.ca/Papers1/43-3/paper43-3-1.pdf">Heptagonal numbers in the associated Pell sequence and Diophantine equations x^2(5x - 3)^2 = 8y^2 ± 4</a>, Fib. Quarterly, 43 (2005), 302-306.

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

%H Claude Soudieux, <a href="/A001333/a001333.pdf">De linfini arithmetique</a>, Zurich, 1960. [Annotated scans of selected pages. Contains many sequences including A1333]

%H E. R. Suryanarayan, <a href="http://www.fq.math.ca/Scanned/36-1/suryanarayan.pdf">The Brahmagupta Polynomials</a>, Fibonacci Quarterly, 34.1 (1996), 30-39.

%H Albert Tarn, <a href="/A001333/a001333_1.pdf">Approximations to certain square roots and the series of numbers connected therewith</a> [Annotated scanned copy]

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

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

%F a(n) = A055642(A125058(n)). - _Reinhard Zumkeller_, Feb 02 2007

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

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

%F G.f.: (1 - x) / (1 - 2*x - x^2) = 1 / (1 - x / (1 - 2*x / (1 + x))). - _Michael Somos_, Sep 02 2012

%F A000129(2n) = 2*A000129(n)*a(n). - _John McNamara_, Oct 30 2002

%F a(n) = (-i)^n * T(n, i), with T(n, x) Chebyshev's polynomials of the first kind A053120 and i^2 = -1.

%F a(n) = a(n-1) + A052542(n-1), n>1. a(n)/A052542(n) converges to sqrt(1/2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003

%F E.g.f.: exp(x)cosh(x*sqrt(2)). - _Paul Barry_, May 08 2003

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

%F For n > 0, a(n)^2 - (1 + (-1)^(n))/2 = Sum_{k=0...n-1}((2k+1)*A001653(n-1-k)); e.g., 17^2 - 1 = 288 = 1*169 + 3*29 + 5*5 + 7*1; 7^2 = 49 = 1*29 + 3*5 + 5*1. - _Charlie Marion_, Jul 18 2003

%F a(n+2) = A078343(n+1) + A048654(n). - _Creighton Dement_, Jan 19 2005

%F a(n) = A000129(n) + A000129(n-1) = A001109(n)/A000129(n) = sqrt(A001110(n)/A000129(n)^2) = ceiling(sqrt(A001108(n))). - _Henry Bottomley_, Apr 18 2000

%F Also the first differences of A000129 (the Pell numbers) because A052937(n) = A000129(n+1) + 1. - _Graeme McRae_, Aug 03 2006

%F a(n) = Sum_{k=0..n} A122542(n,k). - _Philippe Deléham_, Oct 08 2006

%F For another recurrence see A000129.

%F a(n) = Sum_{k=0..n} A098158(n,k)*2^(n-k). - _Philippe Deléham_, Dec 26 2007

%F a(n) = upper left and lower right terms of [1,1; 2,1]^n. - _Gary W. Adamson_, Mar 12 2008

%F E.g.f.: exp(x)*cosh(x*sqrt(2)) = G(0)/2; G(k)=(-1)^k+1/(((3+sqrt(2))^k)-x*(1+sqrt(2))*((17+12*sqrt(2))^k)/(x*(1+sqrt(2))*((3+sqrt(2))^k)-(k+1)/G(k+1)) ; (continued fraction). - _Sergei N. Gladkovskii_, Dec 04 2011

%F For n>=2, a(n)=F_n(2)+F_(n+1)(2), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} binomial(n-i-1,i)x^(n-2*i-1). - _Vladimir Shevelev_, Apr 13 2012

%F a(-n) = (-1)^n * a(n). - _Michael Somos_, Sep 02 2012

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013

%F G.f.: (1+G(0))/(4*x), where G(k)= x*(2*k-1) - 1 + 4*x + x*(2*k-1)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Aug 14 2013

%F Dirichlet g.f.: (PolyLog(s,1-sqrt(2)) + PolyLog(s,1+sqrt(2)))/2. - _Ilya Gutkovskiy_, Jun 26 2016

%e Convergents are 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.

%e The 15 3 X 2 crossword grids, with white squares represented by an o:

%e ooo ooo ooo ooo ooo ooo ooo oo. o.o .oo o.. .o. ..o oo. .oo

%e ooo oo. o.o .oo o.. .o. ..o ooo ooo ooo ooo ooo ooo .oo oo.

%e If p[1]=1, and p[i]=2, (i>1), and if A is 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. - _Milan Janjic_, Apr 29 2010

%e 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 41*x^5 + 99*x^6 + 239*x^7 + 577*x^8 + ...

%p A001333 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else 2*procname(n-1)+procname(n-2) fi end;

%p Digits := 50; A001333 := n-> round((1/2)*(1+sqrt(2))^n);

%p with(numtheory): cf := cfrac (sqrt(2),1000): [seq(nthnumer(cf,i), i=0..50)];

%p A001333:=-(z+1)/(-1+2*z+z**2); # _Simon Plouffe_ in his 1992 dissertation

%p a:= proc(n) local M; M := (Matrix([[2,1],[1,0]])^n); M[2,1]+M[2,2] end: seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 01 2008

%t Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]], {n, 1, 40}], 1, 1] (* _Stefan Steinerberger_, Apr 08 2006 *)

%t Table[((1 - Sqrt[2])^n + (1 + Sqrt[2])^n)/2, {n, 0, 29}] // Simplify (* _Robert G. Wilson v_, May 02 2006 *)

%t a[0] = 1; a[1] = 1; a[n_] := a[n] = 2a[n - 1] + a[n - 2]; Table[a@n, {n, 0, 29}] (* _Robert G. Wilson v_, May 02 2006 *)

%t Table[ MatrixPower[{{1, 2}, {1, 1}}, n][[1, 1]], {n, 0, 30}] (* _Robert G. Wilson v_, May 02 2006 *)

%t a=c=0;t={b=1}; Do[c=a+b+c; AppendTo[t,c]; a=b;b=c,{n,40}]; t (* _Vladimir Joseph Stephan Orlovsky_, Mar 23 2009 *)

%t LinearRecurrence[{2, 1}, {1, 1}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Mar 23 2009 *)

%t Join[{1}, Numerator[Convergents[Sqrt[2], 30]]] (* _Harvey P. Dale_, Aug 22 2011 *)

%t Table[(-I)^n ChebyshevT[n, I], {n, 10}] (* _Eric W. Weisstein_, Apr 04 2017 *)

%t CoefficientList[Series[(-1 + x)/(-1 + 2 x + x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)

%t Table[Sqrt[(ChebyshevT[n, 3] + (-1)^n)/2], {n, 0, 20}] (* _Eric W. Weisstein_, Apr 17 2018 *)

%o (PARI) {a(n) = if( n<0, (-1)^n, 1) * contfracpnqn( vector( abs(n), i, 1 + (i>1))) [1, 1]} /* _Michael Somos_, Sep 02 2012 */

%o (PARI) {a(n) = polchebyshev(n, 1, I) / I^n} /* _Michael Somos_, Sep 02 2012 */

%o (Sage) from sage.combinat.sloane_functions import recur_gen2

%o it = recur_gen2(1,1,2,1)

%o [it.next() for i in range(30)] ## _Zerinvary Lajos_, Jun 24 2008

%o (Sage) [lucas_number2(n,2,-1)/2 for n in xrange(0, 30)] # _Zerinvary Lajos_, Apr 30 2009

%o (PARI) { default(realprecision, 2000); for (n=0, 4000, a=contfracpnqn(vector(n, i, 1+(i>1)))[1, 1]; if (a > 10^(10^3 - 6), break); write("b001333.txt", n, " ", a); ); } \\ _Harry J. Smith_, Jun 12 2009

%o a001333 n = a001333_list !! n

%o a001333_list = 1 : 1 : zipWith (+)

%o a001333_list (map (* 2) \$ tail a001333_list)

%o -- _Reinhard Zumkeller_, Jul 08 2012

%Y For denominators see A000129.

%Y See A040000 for the continued fraction expansion of sqrt(2).

%Y a(n)+a(n+1) = 2 A000129(n+1). 2*a(n) = A002203(n) (companion Pell numbers).

%Y See also A078057 which is the same sequence without the initial 1.

%Y Cf. also A152113.

%Y Row sums of unsigned Chebyshev T-triangle A053120. a(n)= A054458(n, 0) (first column of convolution triangle).

%Y Row sums of A140750, A160756, A135837.

%Y Equals A034182(n-1) + 2 and A084128(n)/2^n. First differences of A052937. Partial sums of A052542. Pairwise sums of A048624. Bisection of A002965.

%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 Second row of the array in A135597.

%Y A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176501, A176502. - _Sameen Ahmed Khan_, Jun 28 2010

%Y Cf. A055099. - _Gary W. Adamson_, Aug 14 2010

%Y Cf. A028859, A001906 / A088305, A033303, A000225, A095263, A003945, A006356, A002478, A214260, A001911 and A000217 for other restricted ternary words. - _Olivier Gérard_, Aug 28 2012

%Y Cf. Triangle A106513 (alternating row sums). - _Wolfdieter Lang_, Oct 05 2014

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

%O 0,3

%A _N. J. A. Sloane_, _R. K. Guy_

%E Chebyshev comments from _Wolfdieter Lang_, Jan 10 2003

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Last modified October 17 12:01 EDT 2018. Contains 316279 sequences. (Running on oeis4.)