A192955 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 2, 7, 18, 41, 84, 161, 294, 519, 894, 1513, 2528, 4185, 6882, 11263, 18370, 29889, 48548, 78761, 127670, 206831, 334942, 542257, 877728, 1420561, 2298914, 3720151, 6019794, 9740729, 15761364, 25502993, 41265318, 66769335, 108035742

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 (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 08 2014: (Start)

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

a(n) - a(n-1) = A192954(n-1). (End)

a(n) = 2*Lucas(n+3) - (n^2+4*n+8). - 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+3]-(n^2+4*n+8), {n, 0, 40}] (* G. C. Greubel, Jul 12 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; 2*(f(n+4)+f(n+2))-(n^2+4*n+8)) \\ G. C. Greubel, Jul 12 2019
(Magma) [2*Lucas(n+3)-(n^2+4*n+8): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) [2*lucas_number2(n+3, 1, -1)-(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) List([0..40], n-> 2*Lucas(1, -1, n+3)[2]-(n^2+4*n+8)); # G. C. Greubel, Jul 12 2019

Crossrefs

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

Sequence in context: A051743 A054111 A295054 * A055503 A077802 A095151

Adjacent sequences: A192952 A192953 A192954 * A192956 A192957 A192958

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved