A215043 a(n) = F(12*n)/(24*L(2*n)), n >= 0, with F = A000045 (Fibonacci) and L = A000032 (Lucas).

0, 2, 276, 34561, 4261992, 524393210, 64499742738, 7933009283134, 975696814205904, 120002796170968643, 14759368609635548580, 1815282342961539780022, 223264968937188026209956, 27459775899111901985784506

Offset: 0

Author

Wolfdieter Lang, Aug 31 2012

Keywords

• nonn
• easy

Comments

24*a(n) is the third example for the Riordan transition matrix introduced in a comment on A078812 (with offset [0,0]). Take there l -> n, n -> 2. See the second formula below.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Index entries for linear recurrences with constant coefficients, signature (144,-2640,6930,-2640,144,-1).

Crossrefs

Cf. A215042 (for F(8*n)/L(2*n)).

Programs

Magma

[Fibonacci(12*n)/(24*Lucas(2*n)): n in [0..15]]; // Vincenzo Librandi, Sep 02 2012

Mathematica

Table[Fibonacci[12*n]/(24*LucasL[2*n]), {n,0,15}] (* G. C. Greubel, Jun 30 2019 *)

PARI

lucas(n) = fibonacci(n+1) + fibonacci(n-1);
vector(15, n, n--; fibonacci(12*n)/(24*lucas(2*n))) \\ G. C. Greubel, Jun 30 2019

Sage

[fibonacci(12*n)/(24*lucas_number2(2*n,1,-1)) for n in (0..15)] # G. C. Greubel, Jun 30 2019

Formula

a(n) = F(12*n)/(24*L(2*n)), n >= 0, with F = A000045 (Fibonacci) and L = A000032 (Lucas).

a(n) = 3*F(2*n) + 20*F(2*n)^3 + 25*F(2*n)^5, n >= 0 (see the comment above).

O.g.f.: x*(2 - 12*x + 97*x^2 - 12*x^3 + 2*x^4)/((1 - 3*x + x^2)*(1 - 18*x + x^2)*(1 - 123*x + x^2)). From the o.g.f.s for the sequences appearing in the preceding formula, see A001906, A215039 and A215044.

a(n) = (L(8*n) + 1)*F(2*n)/24. - Ehren Metcalfe, Jun 04 2019