%I #14 Jul 21 2015 13:05:47
%S 1,2,8,22,68,188,532,1444,3921,10446,27704,72714,189912,492760,
%T 1273064,3273896,8389489,21423994,54550728,138520286,350899964,
%U 886925652,2237284668,5633150988,14159465505,35535456518,89053087224,222870328210,557074041840,1390807477040
%N Modified variant of A006645, the self-convolution of the Pell series.
%C Analogous series using the Fibonacci numbers as a generator = A089098.
%H Colin Barker, <a href="/A178159/b178159.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-12,4,4,4,4,1).
%F (1/2) * [ (1, 4, 14, 44, 131,...) + (1, 0, 2, 0, 5,...)]; where (1, 4, 14, 44,...) = A006645, the self-convolution of the Pell series, and (1, 0, 2, 0, 5,...) = the aerated Pell series.
%F G.f.: -x*(2*x^3-2*x+1) / ((x^2+2*x-1)^2*(x^4+2*x^2-1)). - _Colin Barker_, Jul 21 2015
%e (1/2) * (1, 4, 14, 44, 131,...) + (1, 0, 2, 0, 5,...) = (1/2) * (2, 4, 16, 44, 136, 376,...) = (1, 2, 8, 22, 68, 188,...).
%p A178159 := proc(n)
%p if type (n,'even') then
%p (A006645(n+2) + A000129(n/2+1))/2 ;
%p else
%p A006645(n+2)/2 ;
%p fi;
%p end proc: # _R. J. Mathar_, Jul 21 2015
%o (PARI) Vec(-x*(2*x^3-2*x+1)/((x^2+2*x-1)^2*(x^4+2*x^2-1)) + O(x^40)) \\ _Colin Barker_, Jul 21 2015
%Y Cf. A089098, A006645.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Dec 18 2010
%E Corrected by _R. J. Mathar_, Jul 21 2015