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Column 2 of A007754.
5

%I #34 Jul 30 2023 11:22:31

%S 1,3,11,52,301,2055,16139,143196,1415821,15430835,183754199,

%T 2373373752,33043478329,493278801183,7859417340599,133116815989000,

%U 2388243270461401,45243505322777619,902481863185090979

%N Column 2 of A007754.

%C Conjectured partial sums of A093935. - _Sean A. Irvine_, Jun 25 2022

%H Seiichi Manyama, <a href="/A058799/b058799.txt">Table of n, a(n) for n = 0..448</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModularGroupGamma.html">Modular Group Gamma</a>. [_Roger L. Bagula_, Nov 02 2008]

%F 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

%F 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

%F 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

%t 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 *)

%t nxt[{n_,a_,b_}]:={n+1,b,(n+3)b-a}; NestList[nxt,{0,0,1},20][[;;,2]] (* _Harvey P. Dale_, Jul 30 2023 *)

%Y Cf. A007754, A058798.

%K nonn

%O 0,2

%A _Christian G. Bower_, Dec 02 2000