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 A192970 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
 0, 1, 3, 9, 21, 44, 85, 156, 276, 476, 806, 1347, 2230, 3667, 6001, 9787, 15923, 25862, 41955, 68006, 110170, 178406, 288828, 467509, 756636, 1224469, 1981455, 3206301, 5188161, 8394896, 13583521, 21978912, 35562960, 57542432, 93105986 (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) + n(n+3)/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-x+2*x^2-x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014 a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 7*n + 14)/2. - Ehren Metcalfe, Jul 13 2019 MATHEMATICA (* First progream *) q = x^2; s = x + 1; z = 40; p[0, x]:= 1; p[n_, x_]:= x*p[n-1, x] + n*(n+3)/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}] (* A192969 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192970 *) (* Additional programs *) CoefficientList[Series[x*(1-x+2*x^2-x^3)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 13 2019 *) Table[LucasL[n+3]+Fibonacci[n+4]-(n^2+7*n+14)/2, {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *) PROG (MAGMA) [Fibonacci(n+4)+Lucas(n+3)-(n^2+7*n+14)/2: n in [0..40]]; // Vincenzo Librandi, Jul 13 2019 (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2)-(n^2+7*n+14)/2) \\ G. C. Greubel, Jul 24 2019 (Sage) f=fibonacci; [2*f(n+4)+f(n+2)-(n^2+7*n+14)/2 for n in (0..40)] # G. C. Greubel, Jul 24 2019 (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2)-(n^2+7*n+14)/2); # G. C. Greubel, Jul 24 2019 CROSSREFS Cf. A000032, A000045, A192232, A192744, A192951, A192970. Sequence in context: A080549 A175006 A084569 * A110964 A107351 A068156 Adjacent sequences:  A192967 A192968 A192969 * A192971 A192972 A192973 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 13 2011 STATUS approved

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Last modified August 2 08:34 EDT 2021. Contains 346422 sequences. (Running on oeis4.)