A192954 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 1, 5, 11, 23, 43, 77, 133, 225, 375, 619, 1015, 1657, 2697, 4381, 7107, 11519, 18659, 30213, 48909, 79161, 128111, 207315, 335471, 542833, 878353, 1421237, 2299643, 3720935, 6020635, 9741629, 15762325, 25504017, 41266407, 66770491

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 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 08 2014: (Start)

G.f.: (1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2).

a(n) - a(n-1) = A168674(n-1). (End)

a(n) = 2*Lucas(n+2) - (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;
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}] (* A192954 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
(* Second program *)
Table[2*LucasL[n+2]-(2*n+5), {n, 0, 40}] (* G. C. Greubel, Jul 12 2019 *)
LinearRecurrence[{3, -2, -1, 1}, {1, 1, 5, 11}, 40] (* Harvey P. Dale, Jan 13 2022 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; 2*(f(n+3)+f(n+1))-(2*n+5)) \\ G. C. Greubel, Jul 12 2019
(Magma) [2*Lucas(n+2)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) [2*lucas_number2(n+2, 1, -1)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) List([0..40], n-> 2*Lucas(1, -1, n+2)[2]-(2*n+5)); # G. C. Greubel, Jul 12 2019

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192955.

Sequence in context: A210731 A295959 A156109 * A337437 A107010 A175942

Adjacent sequences: A192951 A192952 A192953 * A192955 A192956 A192957

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved