OFFSET
0,2
COMMENTS
Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(1,4).
a(n) is the number of compositions of n when there are 4 types of ones. - Milan Janjic, Aug 13 2010
a(n)/a(n-1) tends to (5 + sqrt(13))/2 = 4.30277563... . - Gary W. Adamson, Jul 30 2013
a(n) counts closed walks on K_2 containing four loops on the index vertex and one loop on the other. Equivalently the (1,1)_entry of A^(n) where the adjacency matrix of digraph is A=(4,1;1,1). - David Neil McGrath, Nov 05 2014
Number of words of length n over {0,1,...,5} in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory (Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 474
Index entries for linear recurrences with constant coefficients, signature (5,-3).
FORMULA
A member of the family of sequences defined by a(n) = (a(1)+1)*a(n-1) - (a(1)-1)*a(n-2). Alternatively, invert A007052 (invert: define b by 1 + Sum a(n)*x^n = 1/(1 - Sum b(n)*x^n)).
a(n+1)*a(n+1) - a(n+2)*a(n) = -3^n for n>0. - D. G. Rogers, Jul 11 2004
O.g.f.: (1-x)/(1-5*x+3*x^2). - R. J. Mathar, Nov 23 2007
a(n) = 4*a(n-1) + a(n-2) + a(n-3) + a(n-4) + ... + a(0). - Gary W. Adamson, Aug 12 2013
a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-3+sqrt(13)) + (3+sqrt(13))*(5+sqrt(13))^n)) / sqrt(13). - Colin Barker, Jan 20 2017
E.g.f.: exp(5*x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13. - Stefano Spezia, Jul 09 2022
a(n) = Fibonacci(2*n+1) + 2*Sum_{k=0..n-1} a(k)*Fibonacci(2*(n-1-k)+1). - Greg Dresden and Mulong Xu, Aug 10 2024
MATHEMATICA
LinearRecurrence[{5, -3}, {1, 4}, 40] (* Harvey P. Dale, Jan 14 2012 *)
PROG
(Magma) I:=[1, 4]; [n le 2 select I[n] else 5*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 05 2014
(PARI) Vec((1-x) / (1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Jan 20 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved