A085696 a(n) = L(n) * L(n+1) * L(n+2) / 2, where L(n) = Lucas number (A000032).
3, 6, 42, 154, 693, 2871, 12267, 51794, 219678, 930126, 3940797, 16692319, 70711683, 299536446, 1268861682, 5374976354, 22768778133, 96450071031, 408569091147, 1730726388874, 7331474722278, 31056625155606, 131557975542717, 557288527006079, 2360712084085443
Offset: 0
Examples
G.f. = 3 + 6*x + 42*x^2 + 154*x^3 + 693*x^4 + 2871*x^5 + 12267*x^6 + ...
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..1594
- Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).
Programs
-
Mathematica
Array[(Times @@ Map[LucasL, # + {0, 1, 2}])/2 &, 25, 0] (* Michael De Vlieger, Mar 19 2021 *) Times@@#/2&/@Partition[LucasL[Range[0,30]],3,1] (* Harvey P. Dale, Jun 07 2022 *) -
PARI
{a(n) = my(s); n++; s=if(n<0, -1, 1); n=abs(n); s^n * polcoeff( (-1 + 6*x +3*x^2 + 3*x^3) / ((1 + x - x^2) * (1 - 4*x - x^2)) + x * O(x^n), n)}; /* Michael Somos, Sep 17 2014 */
Formula
G.f.: ( 3 - 3 x + 6 x^2 + x^3 )/( 1 - 3 x - 6 x^2 + 3 x^3 + x^4 ).
Recurrence: a(n+4) = 3 a(n+3) + 6 a(n+2) - 3 a(n+1) - a(n).
From Michael Somos, Sep 17 2014: (Start)
a(n) = (6*F(n+1)^3 + 11*F(n+1)^2*F(n) - 3*F(n+1)*F(n)^2 - 2*F(n)^3) / 2 for all n in Z, where F(n) = Fibonacci number (A000045).
a(n) = (2*F(n+1) - F(n)) * (3*F(n+1) + F(n)) * (F(n+1) + 2*F(n)) / 2 for all n in Z.
a(-2-n) = -(-1)^n * a(n) for all n in Z.
0 = a(n)*(-a(n+1) - a(n+2)) + a(n+1)*(-3*a(n+1) + a(n+2)) for all n in Z. (End)
a(n) = -1 + Sum_{k=0..n} L(k)^2 * L(k+1). - Amiram Eldar, Jan 13 2022