A225686 a(n) = Fibonacci(2*n^2), a "Somos-like" sequence.

1, 21, 2584, 2178309, 12586269025, 498454011879264, 135301852344706746049, 251728825683549488150424261, 3210056809456107725247980776292056, 280571172992510140037611932413038677189525

Offset: 1

Author

N. J. A. Sloane, May 23 2013

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..48
• S. B. Ekhad and D. Zeilberger, How To Generate As Many Somos-Like Miracles as You Wish, arXiv:1303.5306 [math.CO], 2013.

Crossrefs

Cf. A000045, A006720, A054783.

Programs

Magma

[Fibonacci(2*n^2): n in [1..10]]; // G. C. Greubel, Aug 09 2018

Maple

A225686 := proc(n)
    if n <= 5 then
        op(n,[1, 21, 2584, 2178309, 12586269025]) ;
    else
        ( 2303*procname(n - 4)*procname(n - 3)*procname(n - 1)
        + 2255*procname(n - 3)^2*procname(n - 2)
        + 329*procname(n - 4)*procname(n - 1)^2
        - 15792*procname(n - 4)*procname(n - 2)^2
        + 329*procname(n - 4)*procname(n - 3)^2
        - 2303*procname(n - 4)^2*procname(n - 2)
        + 441*procname(n - 2)
        - procname(n-2)^3
        -2961*procname(n-4)
        - procname(n-5)*procname(n-2)*procname(n-1)
        + 329*procname(n-5)*procname(n-3)*procname(n-2) )
        / 48/procname(n-4)/procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Jul 09 2013
# second Maple program:
a:= n-> (<<0|1>, <1|1>>^(2*n^2))[1,2]:
seq(a(n), n=1..12);  # Alois P. Heinz, Aug 09 2018

Mathematica

a[ n_] := Fibonacci[2 n^2]; (* Michael Somos, Dec 05 2016 *)

PARI

{a(n) = fibonacci(2 * n^2)}; /* Michael Somos, Dec 05 2016 */

Formula

a(1) = 1, a(2) = 21, a(3) = 2584, a(4) = 2178309, a(5) = 12586269025, and for n>=6, a(n) = ( 2303a(n - 4)a(n - 3)a(n - 1) + 2255a(n - 3)^2 a(n - 2) + 329a(n - 4)a(n - 1)^2 - 15792a(n - 4)a(n - 2)^2 + 329a(n - 4)a(n - 3)^2 - 2303a(n - 4)^2 a(n - 2) + 441a(n - 2) - a(n-2)^3-2961a(n-4) - a(n-5)a(n-2)a(n-1) + 329a(n-5)a(n-3)a(n-2) )/( 48a(n-4)a(n-2) ).

0 = a(n)*([phone]*a(n+4) - 726110*a(n+6)) + a(n+1)*(-76921899285*a(n+3) + 75284537349*a(n+7)) + a(n+2)*([phone]*a(n+2) + 3613692630240*a(n+4) - 1138829425306704*a(n+6) - 34837488*a(n+8)) + a(n+3)*(-527230649330*a(n+3) + 526991761443*a(n+7) + 329*a(n+9)) + a(n+4)*([phone]*a(n+4) - 1636636155*a(n+6) - 4976784*a(n+8)) + a(n+6)*(2255*a(n+6)). for all n in Z. - Michael Somos, Dec 05 2016