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A192963 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
0, 1, 3, 10, 25, 55, 110, 207, 373, 652, 1115, 1877, 3124, 5157, 8463, 13830, 22533, 36635, 59474, 96451, 156305, 253176, 409943, 663625, 1074120, 1738345, 2813115, 4552162, 7366033, 11919007, 19285910, 31205847, 50492749, 81699652 (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 + 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 +3*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014

a(n) = 3*Fibonacci(n+3) + 4*Fibonacci(n+2) - (n^2 + 5*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(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}] (* A192962 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *)

(* Second program *)

With[{F=Fibonacci}, Table[3*F[n+3]+4*F[n+2] -(n^2+5*n+10), {n, 0, 40}]] (* G. C. Greubel, Jul 11 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 3*f(n+4)+4*f(n+2)-(n^2+5*n+10)) \\ G. C. Greubel, Jul 12 2019

(MAGMA) F:=Fibonacci; [3*F(n+4) +4*F(n+2) -(n^2+5*n+10): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(Sage) f=fibonacci; [3*f(n+4) +4*f(n+2) -(n^2+5*n+10) for n in (0..40)] # G. C. Greubel, Jul 12 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 3*F(n+4) +4*F(n+2) -(n^2+5*n+10)); # G. C. Greubel, Jul 12 2019

CROSSREFS

Cf. A000045, A192232, A192744, A192951, A192962.

Sequence in context: A162607 A267574 A047667 * A000247 A097763 A034506

Adjacent sequences:  A192960 A192961 A192962 * A192964 A192965 A192966

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified January 29 10:34 EST 2020. Contains 331337 sequences. (Running on oeis4.)