A058799 Column 2 of A007754.

1, 3, 11, 52, 301, 2055, 16139, 143196, 1415821, 15430835, 183754199, 2373373752, 33043478329, 493278801183, 7859417340599, 133116815989000, 2388243270461401, 45243505322777619, 902481863185090979

Offset: 0

Author

Christian G. Bower, Dec 02 2000

Keywords

• nonn

Comments

Conjectured partial sums of A093935. - Sean A. Irvine, Jun 25 2022

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..448
• Eric Weisstein's World of Mathematics, Modular Group Gamma. [_Roger L. Bagula_, Nov 02 2008]

Crossrefs

Cf. A007754, A058798.

Programs

Mathematica

Clear[S, T, M, v, n]; S = {{0, -1}, {1, 0}}; T = {{1, 1}, {0, 1}}; M[0] = T.S; M[n_] := M[n] = (MatrixPower[T, n].S).M[n - 1]; v[0] = {1, 0}; v[n_] := v[n] = M[n].v[0]; a = Table[v[n][[1]], {n, 0, 30}] (* Roger L. Bagula, Nov 02 2008 *)
nxt[{n_,a_,b_}]:={n+1,b,(n+3)b-a}; NestList[nxt,{0,0,1},20][[;;,2]] (* Harvey P. Dale, Jul 30 2023 *)

Formula

a(n) = (n+2)*a(n-1)-a(n-2) [with a(0)=1 and a(-1)=0] = A058798(n+1) - A058797(n+2). - Henry Bottomley, Feb 28 2001

A signed version with a slightly different start may be obtained from the modular group Gamma: Let S = {{0, -1}, {1, 0}}; T = {{1, 1}, {0, 1}}; m(n) = T^n.S.m(m-1); v(0)={1,0}; v(n)=m(n).v(0); a(n)=v(n)[[1]]. This gives the sequence 1, 0, -1, -3, -11, -52, -301, -2055, -16139, -143196, -1415821, -15430835, ... - Roger L. Bagula, Nov 02 2008

a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*(n-2*k)!*binomial(n-k,k)*binomial(n-k+2,k+2). Cf. A058798. - Peter Bala, Aug 01 2013