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A192971 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 4
1, 2, 9, 21, 44, 83, 149, 258, 437, 729, 1204, 1975, 3225, 5250, 8529, 13837, 22428, 36331, 58829, 95234, 154141, 249457, 403684, 653231, 1057009, 1710338, 2767449, 4477893, 7245452, 11723459, 18969029, 30692610, 49661765, 80354505 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 (3,-2,-1,1).

FORMULA

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

G.f.: (1-x+5*x^2-x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014

a(n) = 4*Fibonacci(n+3) + Lucas(n+2) - 2*(2*n+5). - G. C. Greubel, Jul 24 2019

MATHEMATICA

(* First program *)

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

p[0, x]:= 1;

p[n_, x_]:= x*p[n-1, x] + 2*n^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}] (* A192971 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192972 *)

(* Additional programs *)

With[{F = Fibonacci}, Table[5*F[n+3]+F[n+1] -2*(2*n+5), {n, 0, 40}]] (* G. C. Greubel, Jul 24 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 5*f(n+3)+f(n+1) -2*(2*n+5)) \\ G. C. Greubel, Jul 24 2019

(MAGMA) F:=Fibonacci; [5*F(n+3)+F(n+1) -2*(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(Sage) f=fibonacci; [5*f(n+3)+f(n+1) -2*(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 24 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 5*F(n+3)+F(n+1) -2*(2*n+5)); # G. C. Greubel, Jul 24 2019

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192972.

Sequence in context: A316430 A131476 A023549 * A024850 A237044 A248116

Adjacent sequences:  A192968 A192969 A192970 * A192972 A192973 A192974

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified July 5 15:26 EDT 2020. Contains 335473 sequences. (Running on oeis4.)