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A192979 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments. 3
1, 1, 4, 9, 19, 36, 65, 113, 192, 321, 531, 872, 1425, 2321, 3772, 6121, 9923, 16076, 26033, 42145, 68216, 110401, 178659, 289104, 467809, 756961, 1224820, 1981833, 3206707, 5188596, 8395361, 13584017, 21979440, 35563521, 57543027, 93106616 (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)+1-n+*n^2, 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..35.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

FORMULA

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

G.f.: -(3*x^2-2*x+1) / ((x-1)^2*(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^2 - 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}]   (* A192979 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]   (* A192980 *)

CROSSREFS

Cf. A192232, A192744, A192951, A192980.

Sequence in context: A008113 A008111 A023611 * A232623 A002804 A133649

Adjacent sequences:  A192976 A192977 A192978 * A192980 A192981 A192982

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified December 6 06:58 EST 2016. Contains 278775 sequences.