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 A192982 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
 0, 1, 1, 3, 8, 20, 44, 89, 169, 307, 540, 928, 1568, 2617, 4329, 7115, 11640, 18980, 30876, 50145, 81345, 131851, 213596, 345888, 559968, 906385, 1466929, 2373939, 3841544, 6216212, 10058540, 16275593, 26335033, 42611587, 68947644, 111560320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + (n-1)^2, 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 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1). FORMULA a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5). G.f.: x*(1-3*x+4*x^2)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 12 2014 a(n) = Lucas(n+3) + Fibonacci(n+2) - (n^2 + 2*n + 5). - G. C. Greubel, Jul 25 2019 MATHEMATICA (* First program *) q = x^2; s = x + 1; z = 40; p[0, x]:= 1; p[n_, x_]:= x*p[n-1, x] + (n-1)^2; 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}] (* A192981 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192982 *) (* Additional programs *) CoefficientList[Series[x*(1-3*x+4*x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *) Table[LucasL[n+3]+Fibonacci[n+2]-(n^2+2*n+5), {n, 0, 40}] (* G. C. Greubel, Jul 25 2019 *) PROG (PARI) vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2) -(n^2+2*n+5)) \\ G. C. Greubel, Jul 25 2019 (Magma) F:=Fibonacci; [F(n+4)+2*F(n+2) -(n^2+2*n+5): n in [0..40]]; // G. C. Greubel, Jul 25 2019 (Sage) f=fibonacci; [f(n+4)+2*f(n+2) -(n^2+2*n+5) for n in (0..40)] # G. C. Greubel, Jul 25 2019 (GAP) F:=Fibonacci;; List([0..40], n-> F(n+4)+2*F(n+2) -(n^2+2*n+5)); # G. C. Greubel, Jul 25 2019 CROSSREFS Cf. A000032, A000045, A192232, A192744, A192951, A192981. Sequence in context: A027298 A000236 A109327 * A096585 A057765 A290866 Adjacent sequences: A192979 A192980 A192981 * A192983 A192984 A192985 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 14 2011 STATUS approved

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Last modified September 19 09:52 EDT 2024. Contains 376008 sequences. (Running on oeis4.)