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A192758 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192758 #7 May 04 2014 17:22:41
%S A192758 0,1,2,4,7,13,22,37,61,101,165,269,437,710,1151,1865,3020,4890,7915,
%T A192758 12810,20730,33546,54282,87834,142122,229963,372092,602062,974161,
%U A192758 1576231,2550400,4126639,6677047,10803695,17480751,28284455,45765215
%N A192758 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192758 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+floor((n+4)/4) for n>0, where p(0,x)=1.  For discussions of polynomial reduction, see A192232 and A192744.
%F A192758 Conjecture: G.f.: -x^2 / ( (1+x)*(x^2+1)*(x^2+x-1)*(x-1)^2 ), partial sums of A080239. a(n)-a(n-2) = A097083(n-1). - _R. J. Mathar_, May 04 2014
%t A192758 q = x^2; s = x + 1; z = 40;
%t A192758 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + Floor[(n + 4)/4] /; n > 0;
%t A192758 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192758 reduce[{p1_, q_, s_, x_}] :=
%t A192758 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192758        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192758 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192758 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192758   (* A080239 *)
%t A192758 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192758   (* A192758 *)
%Y A192758 Cf. A192744, A192232.
%K A192758 nonn
%O A192758 1,3
%A A192758 _Clark Kimberling_, Jul 09 2011