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A192754 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 7 07:24 EST 2023. Contains 360112 sequences. (Running on oeis4.)