A192961 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 4, 11, 26, 55, 108, 201, 360, 627, 1070, 1799, 2992, 4937, 8100, 13235, 21562, 35055, 56908, 92289, 149560, 242251, 392254, 634991, 1027776, 1663345, 2691748, 4355771, 7048250, 11404807, 18453900, 29859609, 48314472, 78175107, 126490670

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2 + n^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

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).

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).

From R. J. Mathar, May 09 2014: (Start)

G.f.: x*(1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^3).

a(n) - a(n-1)= A192960(n-1). (End)

a(n) = 2*Fibonacci(n+5) - (n^2 + 4*n + 10). - G. C. Greubel, Jul 12 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^2 + 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}] (* A192960 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192961 *)
(* Second program *)
With[{F=Fibonacci}, Table[2*F[n+5]-(n^2+4*n+10), {n, 0, 40}]] (* G. C. Greubel, Jul 12 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+5)-(n^2+4*n+10)) \\ G. C. Greubel, Jul 12 2019
(Magma) F:=Fibonacci; [2*F(n+5)-(n^2+4*n+10): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) f=fibonacci; [2*f(n+5)-(n^2+4*n+10) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+5)-(n^2+4*n+10)); # G. C. Greubel, Jul 12 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192960.

Sequence in context: A027966 A141534 A320852 * A290989 A027660 A002940

Adjacent sequences: A192958 A192959 A192960 * A192962 A192963 A192964

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved