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
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).
Programs
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Magma
[(Fibonacci(6*n+1)-1)/4: n in [0..20] ]; // Vincenzo Librandi, Aug 23 2011
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Mathematica
Table[(Fibonacci[6n+1]-1)/4,{n,0,20}] (* or *) LinearRecurrence[ {19,-19,1},{0,3,58},20] (* Harvey P. Dale, Aug 22 2011 *) -
PARI
a(n)=fibonacci(6*n+1)>>2 \\ Charles R Greathouse IV, Aug 23 2011
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