A192474 - OEIS

%I #7 Nov 12 2012 10:28:47

%S 3,4,8,17,40,98,247,632,1632,4237,11036,28802,75259,196796,514840,

%T 1347257,3526176,9230050,24161999,63252752,165591088,433512149,

%U 1134931828,2971261442,7778817075,20365132468,53316487592,139584180257,365435810392

%N Constant term of 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*(2*x^4-2*x^3-9*x^2+11*x-3)/((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+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