A192960 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 3, 7, 15, 29, 53, 93, 159, 267, 443, 729, 1193, 1945, 3163, 5135, 8327, 13493, 21853, 35381, 57271, 92691, 150003, 242737, 392785, 635569, 1028403, 1664023, 2692479, 4356557, 7049093, 11405709, 18454863, 29860635, 48315563, 78176265

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).

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

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

a(n) - a(n-1) = A019274(n+2). (End)

a(n) = 2*Fibonacci(n+4) - (2*n + 5). - 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+4]-(2*n+5), {n, 0, 40}]] (* G. C. Greubel, Jul 12 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019
(Magma) F:=Fibonacci; [2*F(n+4)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) f=fibonacci; [2*f(n+4)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)-(2*n+5)); # G. C. Greubel, Jul 12 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192961.

Sequence in context: A322108 A182640 A277643 * A182717 A122768 A344743

Adjacent sequences: A192957 A192958 A192959 * A192961 A192962 A192963

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved