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A192972 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
0, 1, 3, 12, 33, 77, 160, 309, 567, 1004, 1733, 2937, 4912, 8137, 13387, 21916, 35753, 58181, 94512, 153341, 248575, 402716, 652173, 1055857, 1709088, 2766097, 4476435, 7243884, 11721777, 18967229, 30690688, 49659717, 80352327, 130014092 (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) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).

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

a(n) = 4*Fibonacci(n+4) + Lucas(n+3) - 2*(n^2+4*n+8). - 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+4]+F[n+2] -2*(n^2+4*n+8), {n, 0, 40}]] (* G. C. Greubel, Jul 24 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 5*f(n+4)+f(n+2) -2*(n^2+4*n+8)) \\ G. C. Greubel, Jul 24 2019

(MAGMA) F:=Fibonacci; [5*F(n+4)+F(n+2) -2*(n^2+4*n+8): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(Sage) f=fibonacci; [5*f(n+4)+f(n+2) -2*(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 24 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 5*F(n+4)+F(n+2) -2*(n^2+4*n+8)); # G. C. Greubel, Jul 24 2019

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192971.

Sequence in context: A174963 A054602 A083725 * A159228 A054610 A183468

Adjacent sequences:  A192969 A192970 A192971 * A192973 A192974 A192975

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified July 5 16:07 EDT 2020. Contains 335473 sequences. (Running on oeis4.)