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 A192464 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) = 1 + x^n + x^(2n). 4
 2, 4, 7, 16, 38, 95, 242, 624, 1619, 4216, 11002, 28747, 75170, 196652, 514607, 1346880, 3525566, 9229063, 24160402, 63250168, 165586907, 433505384, 1134920882, 2971243731, 7778788418, 20365086100, 53316412567, 139584058864, 365435613974, 956722540271 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Table of n, a(n) for n=1..30. Index entries for linear recurrences with constant coefficients, signature (5,-7,1,3,-1). FORMULA 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 a(n) = 1 - Fibonacci(n) + Fibonacci(1+n) - Fibonacci(2n) + Fibonacci(1+2n). - Friedjof Tellkamp, Nov 22 2021 EXAMPLE The first four polynomials p(n,x) and their reductions are as follows: p(1,x) = 1 + x + x^2 -> 2 + 2x p(2,x) = 1 + x^2 + x^4 -> 4 + 4x p(3,x) = 1 + x^3 + x^6 -> 7 + 10x p(4,x) = 1 + x^4 + x^8 -> 16 + 24x. From these, read A192464 = (2, 4, 7, 16, ...) and 2*A051450 = (2, 4, 10, 24, ...). MATHEMATICA Remove["Global`*"]; q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n); Table[Simplify[p[n, x]], {n, 1, 5}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}] Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192464 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* 2*A051450 *) Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A051450 *) Table[1-Fibonacci[n]+Fibonacci[1+n]-Fibonacci[2n]+Fibonacci[1+2n], {n, 1, 29}] (* Friedjof Tellkamp, Nov 22 2021 *) CROSSREFS Cf. A000045, A192232, A051450. Sequence in context: A319559 A260790 A151378 * A360885 A343869 A137568 Adjacent sequences: A192461 A192462 A192463 * A192465 A192466 A192467 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 01 2011 STATUS approved

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Last modified September 23 09:36 EDT 2023. Contains 365544 sequences. (Running on oeis4.)