%I #30 Nov 17 2023 19:23:26
%S 0,0,1,36,-16,5041728,-19631351040,-62024429150208,
%T -2805793044443561984,-1213280369793911777918976,
%U 6452140445339288271043778576384,-30464666973776461531165746768673505280,2509543205099684468628113981366827179048960,-83207632517142132982462515955707028888811707910062080
%N A Somos-8 sequence.
%D D. G. Cantor (dgc(AT)ccrwest.org), email to N. J. A. Sloane, Nov. 30, 2000.
%H D. G. Cantor, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002211343">On the analogue of the division polynomials for hyperelliptic curves</a>, J. Reine Angew. Math. (Crelle's J.) 447 (1994), pp. 91-145.
%H R. W. Gosper and Richard C. Schroeppel, <a href="http://arxiv.org/abs/math/0703470">Somos Sequence Near-Addition Formulas and Modular Theta Functions</a>, arXiv:math/0703470 [math.NT], 2007.
%H Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita, Takashi Taniguchi, and Yukihiro Uchida, <a href="https://arxiv.org/abs/2310.01013">Periods modulo p of integer sequences associated with division polynomials of genus 2 curves</a>, arXiv:2310.01013 [math.NT], 2023.
%H Alex Stone, <a href="https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/">The Astonishing Behavior of Recursive Sequences</a>, Quanta Magazine, Nov 16 2023, 13 pages.
%F For all n, 0 = u[4] * a[n+4] * a[n-4] + u[3] * a[n+3] * a[n-3] + u[2] * a[n+2] * a[n-2] + u[1] * a[n+1] * a[n-1] + u[0] * a[n]^2, where u[0], ..., u[4] are 314101616640, 25442230947840, 235226865664, -181502208, -16.
%F a(-n) = -a(n) for all n in Z. - _Michael Somos_, Jun 15 2011
%t (* Assuming the first 10 terms are known. *)
%t init = {0, 0, 1, 36, -16, 5041728, -19631351040, -62024429150208, -2805793044443561984, -1213280369793911777918976};
%t init2 = Join[-Rest[init] // Reverse, init]; lg = Length[init];
%t rep = {u[0] -> 314101616640, u[1] -> 25442230947840, u[2] -> 235226865664, u[3] -> -181502208, u[4] -> -16}; Clear[a];
%t rec = u[4] a[n + 4] a[n - 4] + u[3] a[n + 3] a[n - 3] + u[2] a[n + 2] a[n - 2] + u[1] a[n + 1] a[n - 1] + u[0] a[n]^2 /. rep;
%t (* Print[Solve[rec == 0,a[n+4]][[1]] /. n -> n-4]; *)
%t a[n_] := a[n] = (1/a[n - 8])(16(1226959440 a[n - 4]^2 + 99383714640 a[n - 5] a[n - 3] + 918854944 a[n - 6] a[n - 2] - 708993 a[n - 7] a[n - 1]));
%t Do[a[n] = init2[[n + lg]], {n, -(lg - 1), lg - 1}];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 08 2018 *)
%K sign
%O 0,4
%A _N. J. A. Sloane_, Dec 02 2000