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 A192475 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^(n+1)+x^(2n). 2
 2, 5, 11, 26, 63, 157, 398, 1021, 2639, 6854, 17855, 46601, 121770, 318421, 833027, 2179906, 5705471, 14934533, 39094934, 102345101, 267932007, 701437390, 1836358271, 4807602001, 12586390418, 32951476517, 86267889083, 225851947946 (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. LINKS FORMULA 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 a(n) = F(2n) + F(n+1), where F(n) is A000045. - Carl Najafi, May 06 2014 EXAMPLE The first four polynomials p(n,x) and their reductions are as follows: p(1,x)=1+2x^2 -> 3+2x p(2,x)=1+x^3+x^4 -> 4+5x p(3,x)=1+x^4+x^6 -> 8+11x p(4,x)=1+x^5+x^8 -> 17+26x. From these, read A192474=(3,4,8,17,...) and A192475=(2,5,11,26,...) MATHEMATICA q[x_] := x + 1; p[n_, x_] := 1 + x^(n + 1) + 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}]   (* A192474 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192475 *) CROSSREFS Cf. A192232, A192475. Sequence in context: A247471 A082397 A051286 * A192400 A308154 A182053 Adjacent sequences:  A192472 A192473 A192474 * A192476 A192477 A192478 KEYWORD nonn AUTHOR Clark Kimberling, Jul 01 2011 STATUS approved

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Last modified June 19 12:33 EDT 2021. Contains 345128 sequences. (Running on oeis4.)