A192754 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

1, 6, 12, 23, 40, 68, 113, 186, 304, 495, 804, 1304, 2113, 3422, 5540, 8967, 14512, 23484, 38001, 61490, 99496, 160991, 260492, 421488, 681985, 1103478, 1785468, 2888951, 4674424, 7563380, 12237809, 19801194, 32039008, 51840207, 83879220, 135719432

Offset: 0

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn
• easy

Comments

The titular polynomial is defined recursively by p(n,x)=x*p(n-1,x)+5*n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, 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 (2,0,-1).

Crossrefs

Cf. A192744, A192232, A192755.

Programs

Mathematica

p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 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}]
  (* A192754 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192755 *)
LinearRecurrence[{2, 0, -1}, {1, 6, 12}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)

Formula

Conjecture: G.f.: ( 1+4*x ) / ( (x-1)*(x^2+x-1) ), partial sums of A022095. a(n) = A000071(n+3)+4*A000071(n+2). - R. J. Mathar, May 04 2014

a(n) = 8*Fibonacci(n) + 3*Lucas(n) - 5. - Greg Dresden, Oct 10 2020