A084326 a(0)=0, a(1)=1; for n>1, a(n) = 6*a(n-1)-4*a(n-2).

0, 1, 6, 32, 168, 880, 4608, 24128, 126336, 661504, 3463680, 18136064, 94961664, 497225728, 2603507712, 13632143360, 71378829312, 373744402432, 1956951097344, 10246728974336, 53652569456640, 280928500842496, 1470960727228416, 7702050360000512, 40328459251089408

Offset

0,3

Comments

Binomial transform of A001076. - Paul Barry, Aug 25 2003

The ratio a(n+1)/(a(n+1)-4*a(n)) converges to 2 + sqrt(5). - Karl V. Keller, Jr., May 17 2015

References

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Links

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.

Yuhan Jiang, The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model, arXiv:2312.09427 [math.CO], 2023.

Index entries for linear recurrences with constant coefficients, signature (6,-4).

Formula

a(n) = (1/2)*sum(k = 0, n, binomial(n, k)*F(3*k)) where F(k) denotes the k-th Fibonacci number.

a(n) = sqrt(5)((3+sqrt(5))^n - (3-sqrt(5))^n)/10. - Paul Barry, Aug 25 2003

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

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

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

G.f.: x/(1 - 6x + 4x^2). - R. J. Mathar, Sep 09 2008

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

a(n) = 5a(n - 1) + a(n - 2) + a(n - 3) + ... + a(1) + 1. - Gary W. Adamson, Feb 18 2011

a(n) = 2^(n-1)*A001906(n). - R. J. Mathar, Apr 03 2011

Example

a(5) = 6 * a(4) - 4 * a(3) = 6*168 - 4*32 = 880.

Mathematica

Join[{a = 0, b = 1}, Table[c = 6 * b - 4 * a; a = b; b = c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
LinearRecurrence[{6, -4}, {0, 1}, 30] (* Vincenzo Librandi, May 15 2015 *)

Prog

(PARI) a(n)=(1/2)*sum(k=0, n, binomial(n, k)*fibonacci(3*k))
(PARI) a(n)={2^(n-1)*fibonacci(2*n)} \\ Andrew Howroyd, Oct 27 2020
(Sage) [lucas_number1(n, 6, 4) for n in range(0, 22)] # Zerinvary Lajos, Apr 22 2009
(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

Crossrefs

Cf. A030191.

Sequence in context: A046714 A129171 A082585 * A199699 A306900 A137637

Adjacent sequences: A084323 A084324 A084325 * A084327 A084328 A084329

Keyword

nonn,easy

Author

Benoit Cloitre, Jun 21 2003

Status

approved