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%I #69 Dec 21 2023 11:58:55
%S 0,1,6,32,168,880,4608,24128,126336,661504,3463680,18136064,94961664,
%T 497225728,2603507712,13632143360,71378829312,373744402432,
%U 1956951097344,10246728974336,53652569456640,280928500842496,1470960727228416,7702050360000512,40328459251089408
%N a(0)=0, a(1)=1; for n>1, a(n) = 6*a(n-1)-4*a(n-2).
%C Binomial transform of A001076. - _Paul Barry_, Aug 25 2003
%C The ratio a(n+1)/(a(n+1)-4*a(n)) converges to 2 + sqrt(5). - _Karl V. Keller, Jr._, May 17 2015
%D S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H Karl V. Keller, Jr., <a href="/A084326/b084326.txt">Table of n, a(n) for n = 0..1000</a>
%H Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, <a href="https://arxiv.org/abs/2306.14192">alpha-beta-Factorization and the Binary Case of Simon's Congruence</a>, arXiv:2306.14192 [math.CO], 2023.
%H Yuhan Jiang, <a href="https://arxiv.org/abs/2312.09427">The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model</a>, arXiv:2312.09427 [math.CO], 2023.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).
%F a(n) = (1/2)*sum(k = 0, n, binomial(n, k)*F(3*k)) where F(k) denotes the k-th Fibonacci number.
%F a(n) = sqrt(5)((3+sqrt(5))^n - (3-sqrt(5))^n)/10. - _Paul Barry_, Aug 25 2003
%F a(n) = Sum(C(n, 2k+1)5^k 3^(n-2k-1), k = 0, .., Floor[(n-1)/2]). a(n) = 2^(n-1)F(2n). - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
%F a(n) is the rightmost term in M^n * [1 0] where M is the 2X2 matrix [5 1 / 1 1]. The characteristic polynomial of M = x^2 - 6x + 4. a(n)/a(n-1) tends to (3 + sqrt(5)), a root of the polynomial and an eigenvalue of M. - _Gary W. Adamson_, Dec 16 2004
%F a(n) = sum{k = 0..n, sum{j = 0..n, C(n, j)C(j, k)F(j+k)/2}}. - _Paul Barry_, Feb 14 2005
%F G.f.: x/(1 - 6x + 4x^2). - _R. J. Mathar_, Sep 09 2008
%F If p[i] = (4^i-1)/3, 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) = 5a(n - 1) + a(n - 2) + a(n - 3) + ... + a(1) + 1. - _Gary W. Adamson_, Feb 18 2011
%F a(n) = 2^(n-1)*A001906(n). - _R. J. Mathar_, Apr 03 2011
%e a(5) = 6 * a(4) - 4 * a(3) = 6*168 - 4*32 = 880.
%t Join[{a = 0, b = 1}, Table[c = 6 * b - 4 * a; a = b; b = c, {n, 60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)
%t LinearRecurrence[{6, -4}, {0, 1}, 30] (* _Vincenzo Librandi_, May 15 2015 *)
%o (PARI) a(n)=(1/2)*sum(k=0,n,binomial(n,k)*fibonacci(3*k))
%o (PARI) a(n)={2^(n-1)*fibonacci(2*n)} \\ _Andrew Howroyd_, Oct 27 2020
%o (Sage) [lucas_number1(n,6,4) for n in range(0, 22)] # _Zerinvary Lajos_, Apr 22 2009
%o (Magma) [n le 2 select (n-1) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, May 15 2015
%Y Cf. A030191.
%K nonn,easy
%O 0,3
%A _Benoit Cloitre_, Jun 21 2003
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