A192952 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 2, 7, 16, 33, 62, 111, 192, 325, 542, 895, 1468, 2397, 3902, 6339, 10284, 16669, 27002, 43723, 70780, 114561, 185402, 300027, 485496, 785593, 1271162, 2056831, 3328072, 5384985, 8713142, 14098215, 22811448, 36909757, 59721302, 96631159

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 3n - 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.: x*(1 -x +3*x^2)/((1-x-x^2)*(1-x)^2).

a(n) - a(n-1) = A192746(n-2). (End)

a(n) = 4*Fibonacci(n+2) - (3*n+4). - 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] + 3n - 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}] (* A192746 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192952 *)
(* Second program *)
With[{F=Fibonacci}, Table[4*F[n+2]-(3*n+4), {n, 0, 40}]] (* G. C. Greubel, Jul 12 2019 *)

Prog

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

Crossrefs

Cf. A000045, A192232, A192744, A192951.

Sequence in context: A212576 A208995 A023612 * A132738 A259966 A283500

Adjacent sequences: A192949 A192950 A192951 * A192953 A192954 A192955

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved