A192883 Constant term in the reduction by (x^2 -> x + 1) of the polynomial F(n+3)*x^n, where F = A000045 (Fibonacci sequence).

2, 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, 692290561599

Offset

0,1

Comments

See A192872.

a(n) is also the area of the triangle with vertices at (F(n),F(n+1)), (F(n+1),F(n)), and (F(n+3),F(n+4)) where F(n) = A000045(n). - J. M. Bergot, May 22 2014

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Formula

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

a(n) = Fibonacci(n-1) * Fibonacci(n+3). - Gary Detlefs, Oct 19 2011

a(n) = Fibonacci(n+1)^2 + (-1)^n. - Gary Detlefs, Oct 19 2011

G.f.: ( 2-4*x+x^2 ) / ( (1+x)*(1-3*x+x^2) ). - R. J. Mathar, May 07 2014

a(n) = (2^(-1-n)*(7*(-1)^n*2^(1+n) + (3-sqrt(5))^(1+n) + (3+sqrt(5))^(1+n)))/5. - Colin Barker, Sep 29 2016

From Amiram Eldar, Oct 06 2020: (Start)

Sum_{n>=2} 1/a(n) = 7/18.

Sum_{n>=2} (-1)^n/a(n) = (4/phi - 13/6)/3, where phi is the golden ratio (A001622). (End)

a(n) = a(-2-n) for all n in Z. - Michael Somos, Mar 18 2022

Example

G.f. = 2 + 5*x^2 + 8*x^3 + 26*x^4 + 63*x^5 + 170*x^6 + ... - Michael Somos, Mar 18 2022

Maple

with(combinat):seq(fibonacci(n-1)*fibonacci(n+3), n=0..27): # Gary Detlefs, Oct 19 2011

Mathematica

q = x^2; s = x + 1; z = 28;
p[0, x_] := 2; p[1, x_] := 3 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}] (* A192883 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* minus A121646 *)
LinearRecurrence[{2, 2, -1}, {2, 0, 5}, 30] (* G. C. Greubel, Jan 09 2019 *)
a[ n_] := Fibonacci[n+1]^2 + (-1)^n; (* Michael Somos, Mar 18 2022 *)

Prog

(PARI) a(n) = round((2^(-1-n)*(7*(-1)^n*2^(1+n)+(3-sqrt(5))^(1+n)+(3+sqrt(5))^(1+n)))/5) \\ Colin Barker, Sep 29 2016
(PARI) Vec((2+x^2-4*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Sep 29 2016
(PARI) {a(n) = fibonacci(n+1)^2 + (-1)^n}; /* Michael Somos, Mar 18 2022 */
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( ( 2-4*x+x^2)/((1+x)*(1-3*x+x^2)) )); // G. C. Greubel, Jan 09 2019
(Sage) ((2-4*x+x^2 )/((1+x)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 09 2019
(GAP) a:=[2, 0, 5];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 09 2019

Crossrefs

Cf. A001622, A192232, A192744, A192872.

Sequence in context: A011122 A329960 A085009 * A011435 A139309 A264299

Adjacent sequences: A192880 A192881 A192882 * A192884 A192885 A192886

Keyword

nonn,easy

Author

Clark Kimberling, Jul 12 2011

Status

approved