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A192757 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192757 #4 Mar 30 2012 18:57:37
%S A192757 1,3,5,10,17,28,47,76,125,203,329,534,865,1400,2267,3668,5937,9607,
%T A192757 15545,25154,40700,65856,106558,172415,278975,451392,730368,1181762,
%U A192757 1912131,3093895,5006028,8099924,13105954,21205880,34311835,55517717
%N A192757 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192757 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+floor(n*r) for n>0, where p(0,x)=1 and r=(1+sqrt(5))/2.  For discussions of polynomial reduction, see A192232 and A192744.
%t A192757 q = x^2; s = x + 1; z = 40;
%t A192757 p[0, n_] := 1;
%t A192757 p[n_, x_] := x*p[n - 1, x] + Floor[(n + 1)*GoldenRatio] /; n > 0;
%t A192757 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192757 reduce[{p1_, q_, s_, x_}] :=
%t A192757  FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192757        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192757 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192757 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192757 (* A192457 *)
%t A192757 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192757 (* A023611 *)
%Y A192757 Cf. A192744, A192232.
%K A192757 nonn
%O A192757 0,2
%A A192757 _Clark Kimberling_, Jul 09 2011