OFFSET
1,3
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
Seiichi Manyama, Table of n, a(n) for n = 1..387
FORMULA
a(n+1) = [x^n] 1/(1 - n*x - n*x^2). - Paul D. Hanna, Dec 27 2012
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*n^(n-k) for n>=0 (conjectured). - Werner Schulte, Oct 21 2016
a(n) = ((n + sqrt((n-1)*(n+3)) - 1)^n - (n - sqrt((n-1)*(n+3)) - 1)^n) / (2^n * sqrt((n-1)*(n+3))), for n > 1. - Daniel Suteu, Apr 20 2018
a(n) ~ n^(n-1). - Vaclav Kotesovec, Apr 20 2018
a(n+1) = (-sqrt(n)*i)^n * S(n, sqrt(n)*i) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind. - Seiichi Manyama, Feb 28 2021
EXAMPLE
a(4)=45 because if M is the 2 X 2 matrix [0,1;3,3], then M^4 is the 2 X 2 matrix [36,45;135;171].
G.f. = x + x^2 + 6*x^3 + 45*x^4 + 464*x^5 + 6000*x^6 + 93528*x^7 + 1707111*x^8 + ...
MAPLE
with(linalg): a:=proc(n) local A, k: A[1]:=matrix(2, 2, [0, 1, n-1, 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..20);
MATHEMATICA
M[n_] = If[n > 1, MatrixPower[{{0, 1}, {n - 1, n - 1}}, n], {{0, 1}, {1, 1}}]; a = Table[M[n][[1, 2]], {n, 1, 50}]
Table[SeriesCoefficient[1/(1 - n*x - n*x^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 20 2018 *)
PROG
(PARI) {a(n)=polcoeff(1/(1-n*x-n*x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012
(PARI) a(n) = ([0, 1; n-1, n-1]^n)[1, 2]; \\ Michel Marcus, Apr 20 2018
(PARI) a(n) = round((-sqrt(n-1)*I)^(n-1)*polchebyshev(n-1, 2, sqrt(n-1)*I/2)); \\ Seiichi Manyama, Feb 28 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Jun 16 2005
STATUS
approved