A081013 a(n) = Fibonacci(4*n+3) - 2, or Fibonacci(2*n)*Lucas(2*n+3).

0, 11, 87, 608, 4179, 28655, 196416, 1346267, 9227463, 63245984, 433494435, 2971215071, 20365011072, 139583862443, 956722026039, 6557470319840, 44945570212851, 308061521170127, 2111485077978048, 14472334024676219

Offset: 0

Author

R. K. Guy, Mar 01 2003

Keywords

• nonn
• easy

References

• Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A322159.

Programs

GAP

List([0..40], n-> Fibonacci(4*n+3) -2); # G. C. Greubel, Jul 14 2019

Magma

[Fibonacci(4*n+3)-2: n in [0..40]]; // Vincenzo Librandi, Apr 20 2011

Magma

[Fibonacci(4*n+3)-2: n in [0..40]]; // G. C. Greubel, Jul 14 2019

Maple

with(combinat) for n from 0 to 40 do printf(`%d,`,fibonacci(4*n+3)-2) od # James Sellers, Mar 03 2003

Mathematica

LinearRecurrence[{8,-8,1},{0,11,87},40] (* Harvey P. Dale, Dec 05 2013 *)
Table[Fibonacci[2n] LucasL[2n+3], {n,1,40}] (* Rigoberto Florez, Apr 20 2019 *)
Table[Sum[Binomial[2n-1+i, 2n-1-i], {i, 1, 2n-1}]-1, {n, 1, 40}] (* Rigoberto Florez, Apr 20 2019 *)

PARI

my(x='x+O('x^40)); concat([0], Vec(x*(11-x)/((1-x)*(1-7*x+x^2)))) \\ G. C. Greubel, Dec 24 2017

PARI

vector(40, n, n--; fibonacci(4*n+3)-2) \\ G. C. Greubel, Jul 14 2019

Sage

[fibonacci(4*n+3)-2 for n in (0..40)] # G. C. Greubel, Jul 14 2019

Formula

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

From R. J. Mathar, Sep 03 2010: (Start)

G.f.: x*(11 - x)/((1-x)*(1-7*x+x^2)).

a(n) = A033891(n) - 2.

a(n+1) - a(n) = A056914(n+1), n>0. (End)

a(n) = 7*a(n-1) - a(n-2) + 10, n>=2. - R. J. Mathar, Nov 07 2015

From Rigoberto Florez, Apr 20 2019: (Start)

a(n) = Sum_{i=0..2n} F(i)*L(i+2), F(i) = A000045(i) and L(i) = A000032(i).

a(n) = (Sum_{i=1..2n-1} binomial(2n-1+i,2n-1-i)) - 1. (End)

Product_{n>=1} (1 + 1/a(n)) = 2*(1-1/sqrt(5)) = 2*A322159. - Amiram Eldar, Nov 28 2024