A192471 - OEIS

%I #7 Nov 12 2012 10:30:28

%S 2,5,10,24,59,150,386,1001,2606,6800,17767,46458,121538,318045,832418,

%T 2178920,5703875,14931950,39090754,102338337,267921062,701419680,

%U 1836329615,4807555634,12586315394,32951355125,86267692666,225851630136

%N Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+1).

%C For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

%F Empirical G.f.: -x*(2*x-1)*(x^3-3*x^2-x+2)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). [_Colin Barker_, Nov 12 2012]

%e The first four polynomials p(n,x) and their reductions are as follows:

%e p(1,x)=1+x+x^3 -> 2+3x

%e p(2,x)=1+x^2+x^5 -> 5+6x

%e p(3,x)=1+x^3+x^7 -> 10+15x

%e p(4,x)=1+x^4+x^9 -> 24+37x.

%e From these, read

%e A192471=(2,5,10,24,...) and A087124=(3,6,15,37,...)

%t Remove["Global`*"];

%t q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n+1);

%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 (* A192471 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]

%t (* A087124 *)

%Y Cf. A192232.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jul 01 2011