OFFSET
1,2
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).
LINKS
Robert Israel, Table of n, a(n) for n = 1..351
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
EXAMPLE
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].
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*n-2>>^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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Jun 16 2005
STATUS
approved