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%I #14 May 07 2014 00:58:51
%S 2,5,11,26,63,157,398,1021,2639,6854,17855,46601,121770,318421,833027,
%T 2179906,5705471,14934533,39094934,102345101,267932007,701437390,
%U 1836358271,4807602001,12586390418,32951476517,86267889083,225851947946
%N Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^(n+1)+x^(2n).
%C For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F Empirical G.f.: -x*(x^3-3*x^2-3*x+2)/((x^2-3*x+1)*(x^2+x-1)). - _Colin Barker_, Nov 12 2012
%F a(n) = F(2n) + F(n+1), where F(n) is A000045. - _Carl Najafi_, May 06 2014
%e The first four polynomials p(n,x) and their reductions are as follows:
%e p(1,x)=1+2x^2 -> 3+2x
%e p(2,x)=1+x^3+x^4 -> 4+5x
%e p(3,x)=1+x^4+x^6 -> 8+11x
%e p(4,x)=1+x^5+x^8 -> 17+26x.
%e From these, read
%e A192474=(3,4,8,17,...) and A192475=(2,5,11,26,...)
%t q[x_] := x + 1;
%t p[n_, x_] := 1 + x^(n + 1) + x^(2 n);
%t Table[Simplify[p[n, x]], {n, 1, 5}]
%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t (* A192474 *)
%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t (* A192475 *)
%Y Cf. A192232, A192475.
%K nonn
%O 1,1
%A _Clark Kimberling_, Jul 01 2011
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