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Bisection of A005043.
3

%I #30 Jul 26 2020 04:20:40

%S 0,1,6,36,232,1585,11298,83097,625992,4805595,37458330,295673994,

%T 2358641376,18985057351,154000562758,1257643249140,10331450919456,

%U 85317692667643,707854577312178,5897493615536452,49320944483427000,413887836110423787,3484084625456932134,29412628894558563849

%N Bisection of A005043.

%D G. F. Smith, On isotropic tensors and rotation tensors of dimension m and order n, Tensor (N.S.), Vol. 19 (1968), 79-88 (MR0224008).

%H Vincenzo Librandi, <a href="/A099252/b099252.txt">Table of n, a(n) for n = 0..200</a>

%H D. L. Andrews, <a href="/A005043/a005043.pdf">Letter to N. J. A. Sloane</a>, Apr 10 1978.

%F Recurrence: (n+1)*(2*n+1)*a(n) = n*(26*n-7)*a(n-1) - 3*(26*n^2 - 61*n + 39)*a(n-2) + 27*(n-2)*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Oct 17 2012

%F a(n) ~ 3^(2*n+5/2)/(16*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012

%F a(n) = -hypergeom([-2*n - 1, 1/2], [2], 4). - _Peter Luschny_, Jul 26 2020

%p G := (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)): Gser := series(G,x=0,60):

%p seq(coeff(Gser, x^(2*n-1)), n=1..25); # _Emeric Deutsch_

%p a := n -> -hypergeom([-2*n-1, 1/2], [2], 4):

%p seq(simplify(a(n)), n=0..23); # _Peter Luschny_, Jul 26 2020

%t Take[CoefficientList[Series[(1+x-Sqrt[1-2*x-3*x^2])/(2*x*(1+x)), {x, 0, 60}], x], {2, -1, 2}] (* _Vaclav Kotesovec_, Oct 17 2012 *)

%o (PARI) x='x+O('x^66); v=Vec((1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x))); vector(#v\2,n,v[2*n]) \\ _Joerg Arndt_, May 12 2013

%o (Sage)

%o def A():

%o a, b, c, d, n = 0, 1, 1, -1, 1

%o yield 0

%o while True:

%o n += 1

%o a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))

%o c, d = d, (3*(n-1)*c-(2*n-1)*d)//n

%o if n%2: yield -(d + b)*(1-(-1)^n)//2

%o A099252 = A()

%o print([next(A099252) for _ in range(24)]) # _Peter Luschny_, May 16 2016

%Y Cf. A005043, A099251.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Nov 16 2004

%E More terms from _Emeric Deutsch_, Nov 18 2004