A109517 - OEIS

%I #10 Oct 19 2021 21:02:34

%S 1,2,18,252,4880,120750,3639384,129365880,5298720768,245738908890,

%T 12728860100000,728372947109940,45631105330876416,3106354479972026374,

%U 228329428483544787840,18022862954171193750000,1520481402538463932186624,136531862779634547726146994

%N 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)].

%C 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).

%H Robert Israel, <a href="/A109517/b109517.txt">Table of n, a(n) for n = 1..351</a>

%F 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

%e 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].

%p 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);

%p # second Maple program:

%p a:= n-> (<<0|1>, <n-1|2*n-2>>^n)[1, 2]:

%p seq(a(n), n=1..18);  # _Alois P. Heinz_, Oct 19 2021

%t 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}]

%Y Cf. A000045, A000166.

%K nonn

%O 1,2

%A _Roger L. Bagula_, Jun 16 2005