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A193046 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments. 3
1, 1, 17, 83, 275, 727, 1673, 3505, 6873, 12843, 23155, 40639, 69889, 118353, 198097, 328659, 541667, 888311, 1451433, 2365089, 3846201, 6245771, 10131747, 16423103, 26606785, 43088737, 69761873, 112925075, 182770163, 295787863 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The titular polynomials are defined recursively:  p(n,x)=x*p(n-1,x)+n^4, 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

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

FORMULA

a(n) = 5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).

G.f.: (x^5-6*x^4-x^3-21*x^2+4*x-1) / ((x-1)^4*(x^2+x-1)). - Colin Barker, May 11 2014

MATHEMATICA

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

p[0, x] := 1;

p[n_, x_] := x*p[n - 1, x] + n^4;

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}]   (* A193046 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]   (* A193047 *)

CROSSREFS

Cf. A192232, A192744, A192951, A193047.

Sequence in context: A259142 A142059 A318743 * A158528 A197346 A213436

Adjacent sequences:  A193043 A193044 A193045 * A193047 A193048 A193049

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 15 2011

STATUS

approved

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Last modified June 17 19:05 EDT 2019. Contains 324198 sequences. (Running on oeis4.)