A109519 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,-1;n-1,n-1].

-1, -1, -2, -9, -80, -1000, -15336, -276115, -5705728, -133155495, -3464900000, -99490865760, -3125217447936, -106614813012877, -3925516139359360, -155164259295703125, -6553564019985219584, -294562012662334323872, -14038370700094085018112

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

Table of n, a(n) for n=1..19.

Formula

From Seiichi Manyama, Feb 28 2021: (Start)

a(n+1) = [x^n] 1/(-1 + n*x - n*x^2).

a(n+1) = (-1)^(n+1) * Sum_{k=0..n} (-n)^k * binomial(k,n-k).

a(n+1) = (-1) * sqrt(n)^n * S(n, sqrt(n)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind. (End)

Example

a(4)=-9 because if M is the 2 X 2 matrix [0,-1;3,3], then M^4 is the 2 X 2 matrix [ -18,-9,27,9].

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..21);

Mathematica

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

Prog

(Sage) [ -lucas_number1(n+1, n, n) for n in range(0, 19)] # Zerinvary Lajos, Jul 16 2008
(PARI) a(n) = round(-sqrt(n-1)^(n-1)*polchebyshev(n-1, 2, sqrt(n-1)/2)); \\ Seiichi Manyama, Feb 28 2021

Crossrefs

Cf. A000045, A000166, A109516, A109517, A109518.

Sequence in context: A215629 A221460 A122720 * A193208 A279055 A320946

Adjacent sequences: A109516 A109517 A109518 * A109520 A109521 A109522

Keyword

sign

Author

Roger L. Bagula, Jun 16 2005

Status

approved