OFFSET
1,2
COMMENTS
One of the partitions of 16 is 5+5+2+2+1. Convert this Ferrers graph representation to a 5 X 5 binary matrix.
Lim_{n->inf.} = 3.60167913188315425246437..., the characteristic polynomial of m & m^-1 is x^5-3x^4-3x^3+3x^2 and its only positive root is the limit.
LINKS
FORMULA
a(1)=1, a(2)=5, a(3)=16, a(n)= 3a(n-1) + 3a(n-2) - 3a(n-3).
G.f.: -x*(2*x^2-2*x-1) / (3*x^3-3*x^2-3*x+1). - Colin Barker, Oct 19 2013
EXAMPLE
a(4)=60 because m^4.{1,0,0,0,0} = {60, 60, 45, 32, 16} and the first or second element is 60.
MATHEMATICA
a[1] = 1; a[2] = 5; a[3] = 16; a[n_] := a[n] = 3a[n - 1] + 3a[n - 2] - 3a[n - 3]; Table[ a[n], {n, 24}] (* Or *)
m = {{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}}; Table[(MatrixPower[m, n].{1, 0, 0, 0, 0})[[1]], {n, 24}]
PROG
(PARI) Vec(-x*(2*x^2-2*x-1)/(3*x^3-3*x^2-3*x+1) + O(x^100)) \\ Colin Barker, Oct 19 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson and Robert G. Wilson v, Sep 03 2004
STATUS
approved