A192963 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
0, 1, 3, 10, 25, 55, 110, 207, 373, 652, 1115, 1877, 3124, 5157, 8463, 13830, 22533, 36635, 59474, 96451, 156305, 253176, 409943, 663625, 1074120, 1738345, 2813115, 4552162, 7366033, 11919007, 19285910, 31205847, 50492749, 81699652, 132193523, 213894365, 346089148
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
Programs
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GAP
F:=Fibonacci;; List([0..40], n-> 3*F(n+4) +4*F(n+2) -(n^2+5*n+10)); # G. C. Greubel, Jul 12 2019
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Magma
F:=Fibonacci; [3*F(n+4) +4*F(n+2) -(n^2+5*n+10): n in [0..40]]; // G. C. Greubel, Jul 12 2019
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Mathematica
(* First program *) q = x^2; s = x + 1; z = 40; p[0, x]:= 1; p[n_, x_]:= x*p[n-1, x] + n(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}] (* A192962 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *) (* Second program *) With[{F=Fibonacci}, Table[3*F[n+3]+4*F[n+2] -(n^2+5*n+10), {n,0,40}]] (* G. C. Greubel, Jul 11 2019 *) LinearRecurrence[{4,-5,1,2,-1},{0,1,3,10,25},50] (* Harvey P. Dale, Apr 03 2023 *) -
PARI
vector(40, n, n--; f=fibonacci; 3*f(n+4)+4*f(n+2)-(n^2+5*n+10)) \\ G. C. Greubel, Jul 12 2019
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Sage
f=fibonacci; [3*f(n+4) +4*f(n+2) -(n^2+5*n+10) for n in (0..40)] # G. C. Greubel, Jul 12 2019
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 +3*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+3) + 4*Fibonacci(n+2) - (n^2 + 5*n +10). - G. C. Greubel, Jul 12 2019
E.g.f.: 2*exp(x/2)*(25*cosh(sqrt(5)*x/2) + 12*sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x)*(10 + 6*x + x^2). - Stefano Spezia, Aug 30 2025
Comments