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
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Jun 21 2003
STATUS
approved