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 A192953 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments. 3
 0, 1, 2, 6, 13, 26, 48, 85, 146, 246, 409, 674, 1104, 1801, 2930, 4758, 7717, 12506, 20256, 32797, 53090, 85926, 139057, 225026, 364128, 589201, 953378, 1542630, 2496061, 4038746, 6534864, 10573669, 17108594, 27682326, 44790985, 72473378 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The titular polynomials are defined recursively:  p(n,x)=x*p(n-1,x)+2n-1, with p(0,x)=1.  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744. LINKS Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1). FORMULA a(n)=3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: -x*(1-x+2*x^2)/ ((x^2+x-1)*(x-1)^2). - R. J. Mathar, Aug 01 2011 a(n) = -2*n-3+3*A000045(n+2). - R. J. Mathar, Aug 01 2011 a(n) = A131300(n)-1. - R. J. Mathar, Mar 24 2018 MATHEMATICA q = x^2; s = x + 1; z = 40; p[0, x] := 1; p[n_, x_] := x*p[n - 1, x] + 2 n - 1; Table[Expand[p[n, x]], {n, 0, 7}] reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 +        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1] t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}]; u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A111314 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192953 *) CROSSREFS Cf. A192232, A192744, A192951. Sequence in context: A259577 A172348 A254821 * A275970 A124677 A034465 Adjacent sequences:  A192950 A192951 A192952 * A192954 A192955 A192956 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 13 2011 STATUS approved

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Last modified June 25 06:10 EDT 2019. Contains 324347 sequences. (Running on oeis4.)