A049661 a(n) = (Fibonacci(6*n+1) - 1)/4.

0, 3, 58, 1045, 18756, 336567, 6039454, 108373609, 1944685512, 34895965611, 626182695490, 11236392553213, 201628883262348, 3618083506169055, 64923874227780646, 1165011652593882577, 20905285872462105744

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..750 (terms 0..100 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Formula

From R. J. Mathar, Nov 04 2008: (Start)

G.f.: x*(3+x)/((1-x)*(1-18*x+x^2)).

a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). (End)

a(n) = (-1/4+1/40*(9+4*sqrt(5))^(-n)*(5-sqrt(5)+(5+sqrt(5))*(9+4*sqrt(5))^(2*n))). - Colin Barker, Mar 03 2016

Product_{n>=1} (1 - 1/a(n)) = (sqrt(5)+3)/8 = phi^2/4 = cos(Pi/5)^2 = A019863^2 = (A374149 + 1)/10. - Amiram Eldar, Nov 28 2024

Mathematica

Table[(Fibonacci[6n+1]-1)/4, {n, 0, 20}] (* Harvey P. Dale, Aug 22 2011 *)
(* Alternative: *)
LinearRecurrence[ {19, -19, 1}, {0, 3, 58}, 20] (* Harvey P. Dale, Aug 22 2011 *)

Prog

(Magma) [(Fibonacci(6*n+1)-1)/4: n in [0..20] ]; // Vincenzo Librandi, Aug 23 2011
(PARI) a(n)=fibonacci(6*n+1)>>2 \\ Charles R Greathouse IV, Aug 23 2011

Crossrefs

Cf. A000045, A019863, A374149.

Sequence in context: A254653 A243510 A024218 * A295409 A390341 A184950

Adjacent sequences: A049658 A049659 A049660 * A049662 A049663 A049664

Keyword

nonn,easy,changed

Author

Clark Kimberling

Status

approved