|
%I M3499 N1420 #410 Mar 21 2024 08:36:57
%S 0,1,4,15,56,209,780,2911,10864,40545,151316,564719,2107560,7865521,
%T 29354524,109552575,408855776,1525870529,5694626340,21252634831,
%U 79315912984,296011017105,1104728155436,4122901604639,15386878263120,57424611447841,214311567528244
%N a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.
%C 3*a(n)^2 + 1 is a square. Moreover, 3*a(n)^2 + 1 = (2*a(n) - a(n-1))^2.
%C Consecutive terms give nonnegative solutions to x^2 - 4*x*y + y^2 = 1. - _Max Alekseyev_, Dec 12 2012
%C Values y solving the Pellian x^2 - 3*y^2 = 1; corresponding x values given by A001075(n). Moreover, we have a(n) = 2*a(n-1) + A001075(n-1). - _Lekraj Beedassy_, Jul 13 2006
%C Number of spanning trees in 2 X n grid: by examining what happens at the right-hand end we see that a(n) = 3*a(n-1) + 2*a(n-2) + 2*a(n-3) + ... + 2*a(1) + 1, where the final 1 corresponds to the tree ==...=| !. Solving this we get a(n) = 4*a(n-1) - a(n-2).
%C Complexity of 2 X n grid.
%C A016064 also describes triangles whose sides are consecutive integers and in which an inscribed circle has an integer radius. A001353 is exactly and precisely mapped to the integer radii of such inscribed circles, i.e., for each term of A016064, the corresponding term of A001353 gives the radius of the inscribed circle. - _Harvey P. Dale_, Dec 28 2000
%C n such that 3*n^2 = floor(sqrt(3)*n*ceiling(sqrt(3)*n)). - _Benoit Cloitre_, May 10 2003
%C For n>0, ratios a(n+1)/a(n) may be obtained as convergents of the continued fraction expansion of 2+sqrt(3): either as successive convergents of [4;-4] or as odd convergents of [3;1, 2]. - _Lekraj Beedassy_, Sep 19 2003
%C Ways of packing a 3 X (2*n-1) rectangle with dominoes, after attaching an extra square to the end of one of the sides of length 3. With reference to A001835, therefore: a(n) = a(n-1) + A001835(n-1) and A001835(n) = 3*A001835(n-1) + 2*a(n-1). - _Joshua Zucker_ and the Castilleja School Math Club, Oct 28 2003
%C a(n+1) is a Chebyshev transform of 4^n, where the sequence with g.f. G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)). - _Paul Barry_, Oct 25 2004
%C This sequence is prime-free, because a(2n) = a(n) * (a(n+1)-a(n-1)) and a(2n+1) = a(n+1)^2 - a(n)^2 = (a(n+1)+a(n)) * (a(n+1)-a(n)). - _Jianing Song_, Jul 06 2019
%C Numbers such that there is an m with t(n+m) = 3*t(m), where t(n) are the triangular numbers A000217. For instance, t(35) = 3*t(20) = 630, so 35 - 20 = 15 is in the sequence. - _Floor van Lamoen_, Oct 13 2005
%C a(n) = number of distinct matrix products in (A + B + C + D)^n where commutator [A,B] = 0 but neither A nor B commutes with C or D. - _Paul D. Hanna_ and _Max Alekseyev_, Feb 01 2006
%C For n > 1, middle side (or long leg) of primitive Pythagorean triangles having an angle nearing Pi/3 with larger values of sides. [Complete triple (X, Y, Z), X < Y < Z, is given by X = A120892(n), Y = a(n), Z = A120893(n), with recurrence relations X(i+1) = 2*{X(i) - (-1)^i} + a(i); Z(i+1) = 2*{Z(i) + a(i)} - (-1)^i.] - _Lekraj Beedassy_, Jul 13 2006
%C From _Dennis P. Walsh_, Oct 04 2006: (Start)
%C Number of 2 X n simple rectangular mazes. A simple rectangular m X n maze is a graph G with vertex set {0, 1, ..., m} X {0, 1, ..., n} that satisfies the following two properties: (i) G consists of two orthogonal trees; (ii) one tree has a path that sequentially connects (0,0),(0,1), ..., (0,n), (1,n), ...,(m-1,n) and the other tree has a path that sequentially connects (1,0), (2,0), ..., (m,0), (m,1), ..., (m,n). For example, a(2) = 4 because there are four 2 X 2 simple rectangular mazes:
%C __ __ __ __
%C | | | |__ | | | | __|
%C | __| | __| | |__| | __|
%C (End)
%C [1, 4, 15, 56, 209, ...] is the Hankel transform of [1, 1, 5, 26, 139, 758, ...](see A005573). - _Philippe Deléham_, Apr 14 2007
%C The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4, 26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075, denominators=A001353. - _Clark Kimberling_, Aug 27 2008
%C From _Gary W. Adamson_, Jun 21 2009: (Start)
%C A001353 and A001835 = bisection of continued fraction [1, 2, 1, 2, 1, 2, ...], i.e., of [1, 3, 4, 11, 15, 41, ...].
%C For n>0, a(n) equals the determinant of an (n-1) X (n-1) tridiagonal matrix with ones in the super and subdiagonals and (4, 4, 4, ...) as the main diagonal. [Corrected by _Johannes Boot_, Sep 04 2011]
%C A001835 and A001353 = right and next to right borders of triangle A125077. (End)
%C a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 4's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - _John M. Campbell_, Jun 09 2011
%C 2a(n) is the number of n-color compositions of 2n consisting of only even parts; see Guo in references. - _Brian Hopkins_, Jul 19 2011
%C Pisano period lengths: 1, 2, 6, 4, 3, 6, 8, 4, 18, 6, 10, 12, 12, 8, 6, 8, 18, 18, 5, 12, ... - _R. J. Mathar_, Aug 10 2012
%C From _Michel Lagneau_, Jul 08 2014: (Start)
%C a(n) is defined also by the recurrence a(1)=1; for n>1, a(n+1) = 2*a(n) + sqrt(3*a(n)^2 + 1) where a(n) is an integer for every n. This sequence is generalizable by the sequence b(n,m) of parameter m with the initial condition b(1,m) = 1, and for n > 1 b(n+1,m) = m*b(n,m) + sqrt((m^2 - 1)*b(n,m)^2 + 1) for m = 2, 3, 4, ... where b(n,m) is an integer for every n.
%C The first corresponding sequences are
%C b(n,2) = a(n) = A001353(n);
%C b(n,3) = A001109(n);
%C b(n,4) = A001090(n);
%C b(n,5) = A004189(n);
%C b(n,6) = A004191(n);
%C b(n,7) = A007655(n);
%C b(n,8) = A077412(n);
%C b(n,9) = A049660(n);
%C b(n,10) = A075843(n);
%C b(n,11) = A077421(n);
%C ....................
%C We obtain a general sequence of polynomials {b(n,x)} = {1, 2*x, 4*x^2 - 1, 8*x^3 - 4*x, 16*x^4 - 12*x^2 + 1, 32*x^5 - 32*x^3 + 6*x, ...} with x = m where each b(n,x) is a Gegenbauer polynomial defined by the recurrence b(n,x)- 2*x*b(n-1,x) + b(n-2,x) = 0, the same relation as the Chebyshev recurrence, but with the initial conditions b(x,0) = 1 and b(x,1) = 2*x instead b(x,0) = 1 and b(x,1) = x for the Chebyshev polynomials. (End)
%C If a(n) denotes the n-th term of the above sequence and we construct a triangle whose sides are a(n) - 1, a(n) + 1 and sqrt(3a(n)^2 + 1), then, for every n the measure of one of the angles of the triangle so constructed will always be 120 degrees. This result of ours was published in Mathematics Spectrum (2012/2013), Vol. 45, No. 3, pp. 126-128. - _K. S. Bhanu_ and Dr. M. N. Deshpande, Professor (Retd), Department of Statistics, Institute of Science, Nagpur (India).
%C For n >= 1, a(n) equals the number of 01-avoiding words of length n - 1 on alphabet {0, 1, 2, 3}. - _Milan Janjic_, Jan 25 2015
%C For n > 0, 10*a(n) is the number of vertices and roots on level n of the {4, 5} mosaic (see L. Németh Table 1 p. 6). - _Michel Marcus_, Oct 30 2015
%C (2 + sqrt(3))^n = A001075(n) + a(n)*sqrt(3), n >= 0; integers in the quadratic number field Q(sqrt(3)). - _Wolfdieter Lang_, Feb 16 2018
%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_, Dec 12 2019
%C The Cholesky decomposition A = C C* for tridiagonal A with A[i,i] = 4 and A[i+1,i] = A[i,i+1] = -1, as it arises in the discretized 2D Laplace operator (Poisson equation...), has nonzero elements C[i,i] = sqrt(a(i+1)/a(i)) = -1/C[i+1,i], i = 1, 2, 3, ... - _M. F. Hasler_, Mar 12 2021
%C The triples (a(n-1), 2a(n), a(n+1)), n=2,3,..., are exactly the triples (a,b,c) of positive integers a < b < c in arithmetic progression such that a*b+1, b*c+1, and c*a+1 are perfect squares. - _Bernd Mulansky_, Jul 10 2021
%D J. Austin and L. Schneider, Generalized Fibonacci sequences in Pythagorean triple preserving sequences, Fib. Q., 58:1 (2020), 340-350.
%D Bastida, Julio R., Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163-166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009)
%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 163.
%D F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.
%D J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
%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 T. D. Noe, <a href="/A001353/b001353.txt">Table of n, a(n) for n = 0..200</a>
%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H Christian Aebi and Grant Cairns, <a href="https://arxiv.org/abs/2006.07566">Lattice Equable Parallelograms</a>, arXiv:2006.07566 [math.NT], 2020.
%H Christian Aebi and Grant Cairns, <a href="https://arxiv.org/abs/2401.08827">Equable Parallelograms on the Eisenstein Lattice</a>, arXiv:2401.08827 [math.CO], 2024. See p. 14.
%H W. K. Alt, <a href="http://people.reed.edu/~davidp/homepage/students/alt.pdf">Enumeration of Domino Tilings on the Projective Grid Graph</a>, A Thesis Presented to The Division of Mathematics and Natural Sciences, Reed College, May 2013.
%H K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H Francesca Arici and Jens Kaad, <a href="https://arxiv.org/abs/2012.11186">Gysin sequences and SU(2)-symmetries of C*-algebras</a>, arXiv:2012.11186 [math.OA], 2020.
%H Krassimir T. Atanassov and Anthony G. Shannon, <a href="https://doi.org/10.7546/nntdm.2020.26.3.218-223">On intercalated Fibonacci sequences</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 218-223.
%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="http://arxiv.org/abs/1505.06339">Linear recurrence sequences with indices in arithmetic progression and their sums</a>, arXiv preprint arXiv:1505.06339 [math.NT], 2015.
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 12.
%H K. S. Bhanu and M. N. Deshpande, <a href="https://drive.google.com/file/d/19By_22wSILeALFOx3a0ox6bSI42rixgu/view">Integral triangles with 120° angle</a> Mathematics Spectrum, 45 (3) (2012/2013), 126-128.
%H Latham Boyle and 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 Fabrizio Canfora, Maxim Kurkov, Luigi Rosa, and Patrizia Vitale, <a href="http://arxiv.org/abs/1505.06342">The Gribov problem in Noncommutative QED</a>, arXiv preprint arXiv:1505.06342 [hep-th], 2015.
%H Niccolò Castronuovo, <a href="https://arxiv.org/abs/2102.02739">On the number of fixed points of the map gamma</a>, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
%H Z. Cinkir, <a href="http://arxiv.org/abs/1503.06353">Effective Resistances, Kirchhoff index and Admissible Invariants of Ladder Graphs</a>, arXiv preprint arXiv:1503.06353 [math.CO], 2015.
%H J. B. Cosgrave and K. Dilcher, <a href="https://doi.org/10.1090/mcom/3111">A role for generalized Fermat numbers</a>, Math. Comp. 86 (2017), 899-933 (<a href="http://www.johnbcosgrave.com/publications.php">see also paper #10</a>.
%H M. N. Deshpande, <a href="http://dx.doi.org/10.1080/002073902753586337">One Interesting Family of Diophantine Triplets</a>, International Journal of Mathematical Education In Science and Technology, Vol. 33 (No. 2, Mar-Apr), 2002.
%H M. N. Deshpande, Hansruedi Widmer and Zachary Franco, <a href="https://www.jstor.org/stable/2589631">Simultaneous Squares from Arithmetic Progressions: 10622</a>, The American Mathematical Monthly Vol. 106, No. 9 (Nov., 1999), 867-868.
%H Tomislav Doslic, <a href="http://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
%H G. Dresden and Y. Li, <a href="https://arxiv.org/abs/2210.04322">Periodic Weighted Sums of Binomial Coefficients</a>, arXiv:2210.04322 [math.NT], 2022.
%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.
%H F. Faase, <a href="http://www.iwriteiam.nl/Cpaper.zip">On the number of specific spanning subgraphs of the graphs G X P_n</a>, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>
%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 D. Fortin, <a href="http://ijpam.eu/contents/2012-77-1/11/11.pdf">B-spline Toeplitz Inverse Under Corner Perturbations</a>, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From _N. J. A. Sloane_, Oct 22 2012
%H Dale Gerdemann, <a href="https://www.youtube.com/watch?v=xcUGkPgo08k">Fractal images from (4, -1) recursion</a>, YouTube, Oct 27 2014.
%H Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2108.06462">Fibonacci colored compositions and applications</a>, arXiv:2108.06462 [math.CO], 2021.
%H Andrew Granville and Zhi-Wei Sun, <a href="http://projecteuclid.org/euclid.pjm/1102366187">Values of Bernoulli polynomials</a>, Pacific J. Math. 172 (1996), 117-137, at p. 119.
%H T. N. E. Greville, <a href="http://dx.doi.org/10.1090/S0025-5718-1970-0258238-1">Table for third-degree spline interpolations with equally spaced arguments</a>, Math. Comp., 24 (1970), 179-183.
%H Y.-H. Guo, <a href="https://doi.org/10.1007/s12044-010-0005-4">n-Colour even self-inverse compositions</a>, Proc. Indian Acad. Sci. (Math. Sci.), 120 (2010), 27-33.
%H B. Hopkins, <a href="http://www.westga.edu/~integers/a6intproc11/a6intproc11.pdf">Spotted tilings and n-color compositions</a>, INTEGERS 12B (2012/2013), #A6.
%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., 5.5 (1967), 424-434. Case a=0,b=1; p=4, q=-1.
%H W. D. Hoskins, <a href="http://dx.doi.org/10.1090/S0025-5718-1971-0298873-9">Table for third-degree spline interpolation using equi-spaced knots</a>, Math. Comp., 25 (1971), 797-801.
%H Milan 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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Seong Ju Kim, R. Stees, and L. Taalman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Stees/stees4.html">Sequences of Spiral Knot Determinants</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4
%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 Germain Kreweras, <a href="http://dx.doi.org/10.1016/0095-8956(78)90021-7">Complexite et circuits Euleriens dans les sommes tensorielles de graphes</a>, J. Combin. Theory, B 24 (1978), 202-212.
%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 Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=6.
%H Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.
%H Hojoo Lee, <a href="http://www.math.uu.nl/people/beukers/getaltheorie/pen0795.pdf">Problems in elementary number theory</a> Problem I 18.
%H E. Keith Lloyd, <a href="http://www.jstor.org/stable/3619201">The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles</a>, Math. Gaz. vol 81 (1997), 231-243.
%H Dino Lorenzini and Z. Xiang, <a href="http://alpha.math.uga.edu/~lorenz/IntegralPoints.pdf">Integral points on variable separated curves</a>, Preprint 2016.
%H Valcho Milchev and Tsvetelina Karamfilova, <a href="https://arxiv.org/abs/1707.09741">Domino tiling in grid - new dependence</a>, arXiv:1707.09741 [math.HO], 2017.
%H László Németh, <a href="http://arxiv.org/abs/1510.08311">Trees on hyperbolic honeycombs</a>, arXiv:1510.08311 [math.CO], 2015.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Ariel D. Procaccia and Jamie Tucker-Foltz, <a href="http://procaccia.info/wp-content/uploads/2021/07/compact.pdf">Compact Redistricting Plans Have Many Spanning Trees</a>, Harvard Univ. (2021).
%H P. Raff, <a href="http://www.math.rutgers.edu/~praff/span/2/12/index.xml">Analysis of the Number of Spanning Trees of K_2 x P_n</a>. Contains sequence, recurrence, generating function, and more. [From _Paul Raff_, Mar 06 2009]
%H Ryan Stees, <a href="https://commons.lib.jmu.edu/honors201019/84">Sequences of Spiral Knot Determinants</a>, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
%H D. P. Walsh, <a href="http://www.mtsu.edu/~dwalsh/MAZECNT2.pdf">Counting n x 2 Simple Rectangular Mazes</a>
%H F. V. Waugh and M. W. Maxfield, <a href="http://www.jstor.org/stable/2688511">Side-and-diagonal numbers</a>, Math. Mag., 40 (1967), 74-83.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1)
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F G.f.: x/(1-4*x+x^2).
%F a(n) = ((2 + sqrt(3))^n - (2 - sqrt(3))^n)/(2*sqrt(3)).
%F a(n) = sqrt((A001075(n)^2 - 1)/3).
%F a(n) = 2*a(n-1) + sqrt(3*a(n-1)^2 + 1). - _Lekraj Beedassy_, Feb 18 2002
%F a(n) = -a(-n) for all integer n. - _Michael Somos_, Sep 19 2008
%F Limit_{n->infinity} a(n)/a(n-1) = 2 + sqrt(3). - _Gregory V. Richardson_, Oct 06 2002
%F Binomial transform of A002605.
%F E.g.f.: exp(2*x)*sinh(sqrt(3)*x)/sqrt(3).
%F a(n) = S(n-1, 4) = U(n-1, 2); S(-1, x) := 0, Chebyshev's polynomials of the second kind A049310.
%F a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*4^(n - 2*k). - _Paul Barry_, Oct 25 2004
%F a(n) = Sum_{k=0..n-1} binomial(n+k,2*k+1)*2^k. - _Paul Barry_, Nov 30 2004
%F a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3), n>=3. - _Lekraj Beedassy_, Jul 13 2006
%F a(n) = -A106707(n). - _R. J. Mathar_, Jul 07 2006
%F M^n * [1,0] = [A001075(n), A001353(n)], where M = the 2 X 2 matrix [2,3; 1,2]; e.g., a(4) = 56 since M^4 * [1,0] = [97, 56] = [A001075(4), A001353(4)]. - _Gary W. Adamson_, Dec 27 2006
%F Sequence satisfies 1 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v. - _Michael Somos_, Sep 19 2008
%F Rational recurrence: a(n) = (17*a(n-1)*a(n-2) - 4*(a(n-1)^2 + a(n-2)^2))/a(n-3) for n > 3. - _Jaume Oliver Lafont_, Dec 05 2009
%F If p[i] = Fibonacci(2i) 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. - _Milan Janjic_, May 08 2010
%F a(n) = C_{n-1}^{(1)}(2), where C_n^{(m)}(x) is the Gegenbauer polynomial. - _Eric W. Weisstein_, Jul 16 2011
%F a(n) = -i*sin(n*arccos(2))/sqrt(3). - _Eric W. Weisstein_, Jul 16 2011
%F a(n) = sinh(n*arccosh(2))/sqrt(3). - _Eric W. Weisstein_, Jul 16 2011
%F a(n) = b such that Integral_{x=0..Pi/2} (sin(n*x))/(2-cos(x)) dx = c + b*log(2). - _Francesco Daddi_, Aug 02 2011
%F a(n) = sqrt(A098301(n)) = sqrt([A055793 / 3]), base 3 analog of A031150. - _M. F. Hasler_, Jan 16 2012
%F a(n+1) = Sum_{k=0..n} A101950(n,k)*3^k. - _Philippe Deléham_, Feb 10 2012
%F 1, 4, 15, 56, 209, ... = INVERT(INVERT(1, 2, 3, 4, 5, ...)). - _David Callan_, Oct 13 2012
%F Product_{n >= 1} (1 + 1/a(n)) = 1 + sqrt(3). - _Peter Bala_, Dec 23 2012
%F Product_{n >= 2} (1 - 1/a(n)) = 1/4*(1 + sqrt(3)). - _Peter Bala_, Dec 23 2012
%F a(n+1) = (A001834(n) + A001835(n))/2. a(n+1) + a(n) = A001834(n). a(n+1) - a(n) = A001835(n). - _Richard R. Forberg_, Sep 04 2013
%F a(n) = -(-i)^(n+1)*Fibonacci(n, 4*i), i = sqrt(-1). - _G. C. Greubel_, Jun 06 2019
%F a(n)^2 - a(m)^2 = a(n+m) * a(n-m), a(n+2)*a(n-2) = 16*a(n+1)*a(n-1) - 15*a(n)^2, a(n+3)*a(n-2) = 15*a(n+2)*a(n-1) - 14*a(n+1)*a(n) for all integer n, m. - _Michael Somos_, Dec 12 2019
%F a(n) = 2^n*Sum_{k >= n} binomial(2*k,2*n-1)*(1/3)^(k+1). Cf. A102591. - _Peter Bala_, Nov 29 2021
%F a(n) = Sum_{k > 0} (-1)^((k-1)/2)*binomial(2*n, n+k)*(k|12), where (k|12) is the Kronecker symbol. - _Greg Dresden_, Oct 11 2022
%F Sum_{k=0..n} a(k) = (a(n+1) - a(n) - 1)/2. - _Prabha Sivaramannair_, Sep 22 2023
%F a(2n+1) = A001835(n+1) * A001834(n). - _M. Farrokhi D. G._, Oct 15 2023
%e For example, when n = 3:
%e ****
%e .***
%e .***
%e can be packed with dominoes in 4 different ways: 3 in which the top row is tiled with two horizontal dominoes and 1 in which the top row has two vertical and one horizontal domino, as shown below, so a(2) = 4.
%e ---- ---- ---- ||--
%e .||| .--| .|-- .|||
%e .||| .--| .|-- .|||
%e G.f. = x + 4*x^2 + 15*x^3 + 56*x^4 + 209*x^5 + 780*x^6 + 2911*x^7 + 10864*x^8 + ...
%p A001353 := proc(n) option remember; if n <= 1 then n else 4*A001353(n-1)-A001353(n-2); fi; end;
%p A001353:=z/(1-4*z+z**2); # _Simon Plouffe_ in his 1992 dissertation.
%p seq( simplify(ChebyshevU(n-1, 2)), n=0..20); # _G. C. Greubel_, Dec 23 2019
%t a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, 0, 30}] (* _Robert G. Wilson v_, Jan 13 2005 *)
%t Table[GegenbauerC[n-1, 1, 2]], {n, 0, 30}] (* _Zerinvary Lajos_, Jul 14 2009 *)
%t Table[-((I Sin[n ArcCos[2]])/Sqrt[3]), {n, 0, 30}] // FunctionExpand (* _Eric W. Weisstein_, Jul 16 2011 *)
%t Table[Sinh[n ArcCosh[2]]/Sqrt[3], {n, 0, 30}] // FunctionExpand (* _Eric W. Weisstein_, Jul 16 2011 *)
%t Table[ChebyshevU[n-1, 2], {n, 0, 30}] (* _Eric W. Weisstein_, Jul 16 2011 *)
%t a[0]:=0; a[1]:=1; a[n_]:= a[n]= 4a[n-1] - a[n-2]; Table[a[n], {n, 0, 30}] (* _Alonso del Arte_, Jul 19 2011 *)
%t LinearRecurrence[{4, -1}, {0, 1}, 30] (* _Sture Sjöstedt_, Dec 06 2011 *)
%t Round@Table[Fibonacci[2n, Sqrt[2]]/Sqrt[2], {n, 0, 30}] (* _Vladimir Reshetnikov_, Sep 15 2016 *)
%o (PARI) M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=0,30,print1(([1,0,0]*M^i)[2],",")) \\ Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005
%o (PARI) {a(n) = real( (2 + quadgen(12))^n / quadgen(12) )}; /* _Michael Somos_, Sep 19 2008 */
%o (PARI) {a(n) = polchebyshev(n-1, 2, 2)}; /* _Michael Somos_, Sep 19 2008 */
%o (PARI) concat(0, Vec(x/(1-4*x+x^2) + O(x^30))) \\ _Altug Alkan_, Oct 30 2015
%o (Sage) [lucas_number1(n,4,1) for n in range(30)] # _Zerinvary Lajos_, Apr 22 2009
%o (Sage) [chebyshev_U(n-1,2) for n in (0..20)] # _G. C. Greubel_, Dec 23 2019
%o (Haskell)
%o a001353 n = a001353_list !! n
%o a001353_list =
%o 0 : 1 : zipWith (-) (map (4 *) $ tail a001353_list) a001353_list
%o -- _Reinhard Zumkeller_, Aug 14 2011
%o (GAP) a:=[0,1];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # _Muniru A Asiru_, Feb 16 2018
%o (Magma) I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jun 06 2019
%o (Python)
%o a001353 = [0, 1]
%o for n in range(30): a001353.append(4*a001353[-1] - a001353[-2])
%o print(a001353) # _Gennady Eremin_, Feb 05 2022
%Y Cf. A001075, A001542, A001571, A001834, A001835, A002531, A003500, A005246, A016064, A079935, A082840.
%Y A bisection of A002530.
%Y Cf. A125077.
%Y A row of A116469.
%Y Chebyshev sequence U(n, m): A000027 (m=1), this sequence (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
%Y Cf. A323182.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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