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

%I A225686 #24 Sep 08 2022 08:46:05
%S A225686 1,21,2584,2178309,12586269025,498454011879264,135301852344706746049,
%T A225686 251728825683549488150424261,3210056809456107725247980776292056,
%U A225686 280571172992510140037611932413038677189525
%N A225686 a(n) = Fibonacci(2*n^2), a "Somos-like" sequence.
%H A225686 Seiichi Manyama, <a href="/A225686/b225686.txt">Table of n, a(n) for n = 1..48</a>
%H A225686 S. B. Ekhad and D. Zeilberger, <a href="https://arxiv.org/abs/1303.5306">How To Generate As Many Somos-Like Miracles as You Wish</a>, arXiv:1303.5306 [math.CO], 2013.
%F A225686 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) ).
%F A225686 0 = a(n)*(+233805165*a(n+4) - 726110*a(n+6)) + a(n+1)*(-76921899285*a(n+3) + 75284537349*a(n+7)) + a(n+2)*(+11222647920*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)*(+516007999155*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
%p A225686 A225686 := proc(n)
%p A225686     if n <= 5 then
%p A225686         op(n,[1, 21, 2584, 2178309, 12586269025]) ;
%p A225686     else
%p A225686         ( 2303*procname(n - 4)*procname(n - 3)*procname(n - 1)
%p A225686         + 2255*procname(n - 3)^2*procname(n - 2)
%p A225686         + 329*procname(n - 4)*procname(n - 1)^2
%p A225686         - 15792*procname(n - 4)*procname(n - 2)^2
%p A225686         + 329*procname(n - 4)*procname(n - 3)^2
%p A225686         - 2303*procname(n - 4)^2*procname(n - 2)
%p A225686         + 441*procname(n - 2)
%p A225686         - procname(n-2)^3
%p A225686         -2961*procname(n-4)
%p A225686         - procname(n-5)*procname(n-2)*procname(n-1)
%p A225686         + 329*procname(n-5)*procname(n-3)*procname(n-2) )
%p A225686         / 48/procname(n-4)/procname(n-2) ;
%p A225686     end if;
%p A225686 end proc: # _R. J. Mathar_, Jul 09 2013
%p A225686 # second Maple program:
%p A225686 a:= n-> (<<0|1>, <1|1>>^(2*n^2))[1,2]:
%p A225686 seq(a(n), n=1..12);  # _Alois P. Heinz_, Aug 09 2018
%t A225686 a[ n_] := Fibonacci[2 n^2]; (* _Michael Somos_, Dec 05 2016 *)
%o A225686 (PARI) {a(n) = fibonacci(2 * n^2)}; /* _Michael Somos_, Dec 05 2016 */
%o A225686 (Magma) [Fibonacci(2*n^2): n in [1..10]]; // _G. C. Greubel_, Aug 09 2018
%Y A225686 Cf. A000045, A006720, A054783.
%K A225686 nonn
%O A225686 1,2
%A A225686 _N. J. A. Sloane_, May 23 2013