A109517 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,2(n-1)].

1, 2, 18, 252, 4880, 120750, 3639384, 129365880, 5298720768, 245738908890, 12728860100000, 728372947109940, 45631105330876416, 3106354479972026374, 228329428483544787840, 18022862954171193750000, 1520481402538463932186624, 136531862779634547726146994

Offset: 1

Author

Roger L. Bagula, Jun 16 2005

Keywords

• nonn

Comments

The (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;1,1] is the Fibonacci number A000045(n).

Examples

			a(4)=252 because if M is the 2 X 2 matrix [0,1;3,6], then M^4 is the 2 X 2 matrix [117,252;756;1629].
		

Links

• Robert Israel, Table of n, a(n) for n = 1..351

Crossrefs

Cf. A000045, A000166.

Programs

Maple

with(linalg): a:=proc(n) local A,k: A[1]:=matrix(2,2,[0,1,n-1,2*(n-1)]): for k from 2 to n do A[k]:=multiply(A[k-1],A[1]) od: A[n][1,2] end: seq(a(n),n=1..19);
# second Maple program:
a:= n-> (<<0|1>, >^n)[1, 2]:
seq(a(n), n=1..18);  # Alois P. Heinz, Oct 19 2021

Mathematica

M[n_] = If[n > 1, MatrixPower[{{0, 1}, {n - 1, 2*(n - 1)}}, n], {{0, 1}, {1, 1}}] a = Table[M[n][[1, 2]], {n, 1, 50}]

Formula

For n > 1, a(n) = ((n - 1 + sqrt(n*(n - 1)))^n - (n - 1 - sqrt(n*(n - 1)))^n)/(2*sqrt(n*(n - 1))). - Robert Israel, Oct 19 2021