A098347 A sequence derived from a Ferrers graph partition of 16.

1, 5, 16, 60, 213, 771, 2772, 9990, 35973, 129573, 466668, 1680804, 6053697, 21803499, 78529176, 282836934, 1018687833, 3668986773, 13214513016, 47594435868, 171419886333, 617399427555, 2223674634060, 8008962525846

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

Table of n, a(n) for n=1..24.

Index entries for linear recurrences with constant coefficients, signature (3,3,-3).

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

Sequence in context: A226897 A077235 A203232 * A203414 A189450 A180719

Adjacent sequences: A098344 A098345 A098346 * A098348 A098349 A098350

Keyword

nonn,easy,changed

Author

Gary W. Adamson and Robert G. Wilson v, Sep 03 2004

Status

approved