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A241594 a(n) = -(8*a(n-4)*a(n-1)+57*a(n-3)*a(n-2))/a(n-5) with initial values 1, 0, -1, 1, 8, 57, -455.

%I A241594 #39 Mar 03 2025 13:26:07
%S A241594 1,0,-1,1,8,57,-455,22352,47767,69739671,3385862936,1747973613295,
%T A241594 -632038062613231,319807929289790304,778756000716629557903,
%U A241594 186509371006506937278833,-7581885296067966478838810840,-17592286469464275110206466526327,594744699237794019378328459208828297
%N A241594 a(n) = -(8*a(n-4)*a(n-1)+57*a(n-3)*a(n-2))/a(n-5) with initial values 1, 0, -1, 1, 8, 57, -455.
%H A241594 G. C. Greubel, <a href="/A241594/b241594.txt">Table of n, a(n) for n = 0..90</a>
%H A241594 R. W. Gosper and Richard C. Schroeppel, <a href="https://arxiv.org/abs/math/0703470">Somos Sequence Near-Addition Formulas and Modular Theta Functions</a>, arXiv:math/0703470 [math.NT], 2007. See h_n.
%F A241594 -a(n+1) = A360381(2*n)*(-1)^(n%4!=0). a(n+2)*a(n-2) = -(-1)^n*a(n+1)*a(n-1) + 8*a(n)^2 for all n in Z. - _Michael Somos_, Mar 01 2025
%p A241594 f:=proc(n) option remember;
%p A241594 if n <= 2 then 1-n
%p A241594 elif n=3 then 1
%p A241594 elif n=4 then 8
%p A241594 elif n=5 then 57
%p A241594 elif n=6 then -455
%p A241594 else -(8*f(n-4)*f(n-1)+57*f(n-3)*f(n-2))/f(n-5); fi; end;
%p A241594 [seq(f(n),n=0..30)];
%t A241594 a[n_] := a[n] = If[n <= 6, {1, 0, -1, 1, 8, 57, -455}[[n + 1]], -(8*a[n - 4]*a[n - 1] + 57*a[n - 3]*a[n - 2])/a[n - 5]];
%t A241594 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Dec 02 2017 *)
%t A241594 nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,-(8b e+57c d)/a}; Join[{1,0},NestList[nxt,{-1,1,8,57,-455},20] [[;;,1]]] (* _Harvey P. Dale_, Aug 14 2023 *)
%o A241594 (Magma) I:=[22352, 47767, 69739671, 3385862936, 1747973613295]; [1, 0, -1, 1, 8, 57, -455] cat [n le 5 select I[n] else -(8*Self(n-4)*Self(n-1) + 57*Self(n-3)*Self(n-2))/Self(n-5): n in [1..20]]; // _G. C. Greubel_, Aug 08 2018
%o A241594 (PARI) {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); n--; subst(elldivpol(E, n), 'x, 1) * (-1)^(n%4!=2)}; /* _Michael Somos_, Mar 01 2025 */
%o A241594 (PARI) {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); n--; subst(elldivpol(E, 2*n), 'x, 2) * (-1)^(n + (n%4==2))/6^(n^2 + 1)}; /* _Michael Somos_, Mar 02 2025 */
%Y A241594 See A241595 for another version.
%Y A241594 Cf. A360381.
%K A241594 sign
%O A241594 0,5
%A A241594 _N. J. A. Sloane_, May 18 2014