A192957 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 1, 5, 14, 34, 72, 141, 261, 465, 806, 1370, 2296, 3809, 6273, 10277, 16774, 27306, 44368, 71997, 116725, 189121, 306286, 495890, 802704, 1299169, 2102497, 3402341, 5505566, 8908690, 14415096, 23324685, 37740741, 61066449, 98808278

Offset

0,4

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) +- 1 + 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).

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

G.f.: x*(1 -3*x +6*x^2 -2*x^3)/((1-x-x^2)*(1-x)^3).

a(n) - a(n-1) = A192956(n-1). (End)

a(n) = Fibonacci(n+4) + 4*Fibonacci(n+2) - (n^2+4*n+7). - 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 - 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}] (* A192956 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192957 *)
(* Second program *)With[{F=Fibonacci}, Table[F[n+4]+4*F[n+2]-(n^2+4*n+7), {n, 0, 40}]] (* G. C. Greubel, Jul 12 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; f(n+4)+4*f(n+2)-(n^2+4*n+7)) \\ G. C. Greubel, Jul 12 2019
(Magma) F:=Fibonacci; [F(n+4)+4*F(n+2)-(n^2+4*n+7): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) f=fibonacci; [f(n+4)+4*f(n+2)-(n^2+4*n+7) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4)+4*F(n+2)-(n^2+4*n+7)); # G. C. Greubel, Jul 12 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192956.

Sequence in context: A059821 A296010 A182738 * A094584 A023515 A047860

Adjacent sequences: A192954 A192955 A192956 * A192958 A192959 A192960

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved