A192920 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).

0, 5, 8, 26, 63, 170, 440, 1157, 3024, 7922, 20735, 54290, 142128, 372101, 974168, 2550410, 6677055, 17480762, 45765224, 119814917, 313679520, 821223650, 2149991423, 5628750626, 14736260448, 38580030725, 101003831720, 264431464442

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn

Comments

See A192872.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A000045, A192232, A192744, A192872, A192883, A192919.

Programs

GAP

a:=[0,5,8];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Feb 06 2019

Magma

[Fibonacci(n+2)^2 -(-1)^n: n in [0..30]]; // G. C. Greubel, Feb 06 2019, modified Jul 28 2019

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 3; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^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}] (* A192919 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192920 *)
(* Second program *)
LinearRecurrence[{2,2,-1}, {0,5,8}, 30] (* G. C. Greubel, Feb 06 2019 *)

PARI

vector(30, n, n--; fibonacci(n+2)^2 -(-1)^n) \\ G. C. Greubel, Feb 06 2019, modified Jul 28 2019

Sage

[fibonacci(n+2)^2 -(-1)^n for n in (0..30)] # G. C. Greubel, Feb 06 2019, modified Jul 28 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).

a(n) = A192883(n+1).

G.f.: x*(5-2*x)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, Aug 01 2011

a(n) = (A005248(n+2) - 7*(-1)^n)/5. - R. J. Mathar, Aug 01 2011

a(n) = Fibonacci(n+2)^2 - (-1)^n. - G. C. Greubel, Feb 06 2019

Sum_{n>=1} 1/a(n) = 7/18. - Amiram Eldar, Oct 05 2020