A192973 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 3, 10, 23, 47, 88, 157, 271, 458, 763, 1259, 2064, 3369, 5483, 8906, 14447, 23415, 37928, 61413, 99415, 160906, 260403, 421395, 681888, 1103377, 1785363, 2888842, 4674311, 7563263, 12237688, 19801069, 32038879, 51840074, 83879083

Offset

1,2

Comments

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

Vincenzo Librandi, Table of n, a(n) for n = 1..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.: x*(1+3*x^2)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014

a(n) = Lucas(n+4) - Fibonacci(n-1) - 2*(2*n+3). - Ehren Metcalfe, Jul 13 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 +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}] (* A192973 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192974 *)
(* Additional programs *)
LinearRecurrence[{3, -2, -1, 1}, {1, 3, 10, 23}, 50] (* Vincenzo Librandi, Jul 14 2019 *)
With[{F = Fibonacci}, Table[F[n+4]+3*F[n+2] -2*(2*n+3), {n, 40}]] (* G. C. Greubel, Jul 24 2019 *)

Prog

(Magma) [Lucas(n+4)-Fibonacci(n-1)-2*(2*n+3): n in [1..40]]; // Vincenzo Librandi, Jul 14 2019
(PARI) vector(40, n, f=fibonacci; f(n+4)+3*f(n+2) -2*(2*n+3)) \\ G. C. Greubel, Jul 24 2019
(Sage) f=fibonacci; [f(n+4)+3*f(n+2) -2*(2*n+3) for n in (1..40)] # G. C. Greubel, Jul 24 2019
(GAP) F:=Fibonacci;; List([1..40], n-> F(n+4)+3*F(n+2) -2*(2*n+3)); # G. C. Greubel, Jul 24 2019

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192974.

Sequence in context: A145069 A293350 A256525 * A294503 A080204 A115982

Adjacent sequences: A192970 A192971 A192972 * A192974 A192975 A192976

Keyword

nonn,easy

Author

Clark Kimberling, Jul 13 2011

Status

approved