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

0,2

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).

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

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 *)

Crossrefs

Cf. A192744, A192232, A192755.

Sequence in context: A144568 A222001 A078472 * A005694 A309359 A172079

Adjacent sequences: A192751 A192752 A192753 * A192755 A192756 A192757

Keyword

nonn,easy

Author

Clark Kimberling, Jul 09 2011

Status

approved