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A192748 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
0, 1, 4, 11, 24, 47, 86, 151, 258, 433, 718, 1181, 1932, 3149, 5120, 8311, 13476, 21835, 35362, 57251, 92670, 149981, 242714, 392761, 635544, 1028377, 1663996, 2692451, 4356528, 7049063, 11405678, 18454831, 29860602, 48315529, 78176230 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+3n for n>0, where p(0,x)=1.  For discussions of polynomial reduction, see A192232 and A192744.

LINKS

Table of n, a(n) for n=1..35.

FORMULA

Conjecture: G.f.: -x^2*(1+x+x^2) / ( (x^2+x-1)*(x-1)^2 ), so the first differences are in A154691. - R. J. Mathar, May 04 2014

MATHEMATICA

q = x^2; s = x + 1; z = 40;

p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 3 n;

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}]

  (* A154691 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

  (* A192748 *)

CROSSREFS

Cf. A192744, A192232.

Sequence in context: A057304 A001752 A160860 * A143075 A290707 A322618

Adjacent sequences:  A192745 A192746 A192747 * A192749 A192750 A192751

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 09 2011

STATUS

approved

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Last modified December 12 01:57 EST 2019. Contains 329948 sequences. (Running on oeis4.)