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A192759 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192759 #6 May 04 2014 17:19:49
%S A192759 0,1,2,4,7,12,21,35,58,95,155,253,411,667,1081,1751,2836,4591,7431,
%T A192759 12026,19461,31492,50958,82455,133418,215878,349302,565186,914494,
%U A192759 1479686,2394186,3873879,6268072,10141958,16410037,26552002,42962047
%N A192759 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192759 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+floor((n+5)/5) for n>0, where p(0,x)=1.  For discussions of polynomial reduction, see A192232 and A192744.
%F A192759 Conjecture: G.f.: -x / ( (x^2+x-1)*(x^4+x^3+x^2+x+1)*(x-1)^2 ), partial sums of A124502. - _R. J. Mathar_, May 04 2014
%t A192759 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + Floor[(n + 5)/5] /; n > 0;
%t A192759 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192759 reduce[{p1_, q_, s_, x_}] :=
%t A192759 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192759        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192759 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192759 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192759 (* A124502 *)
%t A192759 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192759 (* A192759 *)
%Y A192759 Cf. A192744, A192232.
%K A192759 nonn
%O A192759 0,3
%A A192759 _Clark Kimberling_, Jul 09 2011