A192464 - OEIS

%I #24 Nov 23 2021 02:28:32

%S 2,4,7,16,38,95,242,624,1619,4216,11002,28747,75170,196652,514607,

%T 1346880,3525566,9229063,24160402,63250168,165586907,433505384,

%U 1134920882,2971243731,7778788418,20365086100,53316412567,139584058864,365435613974,956722540271

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

%C For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.  The coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) = 1 + x^n + x^(2n) is 2*A051450.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,1,3,-1).

%F G.f.: -x*(3*x^4-7*x^3-x^2+6*x-2)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). - _Colin Barker_, Nov 12 2012

%F a(n) = 1 - Fibonacci(n) + Fibonacci(1+n) - Fibonacci(2n) + Fibonacci(1+2n). - _Friedjof Tellkamp_, Nov 22 2021

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

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

%e p(2,x) = 1 + x^2 + x^4 ->  4 +  4x

%e p(3,x) = 1 + x^3 + x^6 ->  7 + 10x

%e p(4,x) = 1 + x^4 + x^8 -> 16 + 24x.

%e From these, read

%e A192464 = (2, 4, 7, 16, ...) and 2*A051450 = (2, 4, 10, 24, ...).

%t Remove["Global`*"];

%t q[x_] := x + 1; p[n_, x_] := 1 + x^n + 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 (* A192464 *)

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

%t (* 2*A051450 *)

%t Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]

%t (* A051450 *)

%t Table[1-Fibonacci[n]+Fibonacci[1+n]-Fibonacci[2n]+Fibonacci[1+2n], {n, 1, 29}]

%t (* _Friedjof Tellkamp_, Nov 22 2021 *)

%Y Cf. A000045, A192232, A051450.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jul 01 2011