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A192961 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
0, 1, 4, 11, 26, 55, 108, 201, 360, 627, 1070, 1799, 2992, 4937, 8100, 13235, 21562, 35055, 56908, 92289, 149560, 242251, 392254, 634991, 1027776, 1663345, 2691748, 4355771, 7048250, 11404807, 18453900, 29859609, 48314472, 78175107, 126490670 (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) + 2 + 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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

FORMULA

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

From R. J. Mathar, May 09 2014: (Start)

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

a(n) - a(n-1)= A192960(n-1). (End)

a(n) = 2*Fibonacci(n+5) - (n^2 + 4*n + 10). - G. C. Greubel, Jul 12 2019

MATHEMATICA

(* First program *)

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

p[0, x]:= 1;

p[n_, x_]:= x*p[n-1, x] + n^2 + 2;

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}] (* A192960 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192961 *)

(* Second program *)

With[{F=Fibonacci}, Table[2*F[n+5]-(n^2+4*n+10), {n, 0, 40}]] (* G. C. Greubel, Jul 12 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+5)-(n^2+4*n+10)) \\ G. C. Greubel, Jul 12 2019

(MAGMA) F:=Fibonacci; [2*F(n+5)-(n^2+4*n+10): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(Sage) f=fibonacci; [2*f(n+5)-(n^2+4*n+10) for n in (0..40)] # G. C. Greubel, Jul 12 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+5)-(n^2+4*n+10)); # G. C. Greubel, Jul 12 2019

CROSSREFS

Cf. A000045, A192232, A192744, A192951, A192960.

Sequence in context: A027966 A141534 A320852 * A290989 A027660 A002940

Adjacent sequences:  A192958 A192959 A192960 * A192962 A192963 A192964

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified August 6 10:14 EDT 2020. Contains 336245 sequences. (Running on oeis4.)