A058231 A Somos-8 sequence.

0, 0, 1, 36, -16, 5041728, -19631351040, -62024429150208, -2805793044443561984, -1213280369793911777918976, 6452140445339288271043778576384, -30464666973776461531165746768673505280, 2509543205099684468628113981366827179048960, -83207632517142132982462515955707028888811707910062080

Offset

0,4

References

D. G. Cantor (dgc(AT)ccrwest.org), email to N. J. A. Sloane, Nov. 30, 2000.

Links

Table of n, a(n) for n=0..13.

D. G. Cantor, On the analogue of the division polynomials for hyperelliptic curves, J. Reine Angew. Math. (Crelle's J.) 447 (1994), pp. 91-145.

R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.

Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita, Takashi Taniguchi, and Yukihiro Uchida, Periods modulo p of integer sequences associated with division polynomials of genus 2 curves, arXiv:2310.01013 [math.NT], 2023.

Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.

Formula

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.

a(-n) = -a(n) for all n in Z. - Michael Somos, Jun 15 2011

Mathematica

(* Assuming the first 10 terms are known. *)
init = {0, 0, 1, 36, -16, 5041728, -19631351040, -62024429150208, -2805793044443561984, -1213280369793911777918976};
init2 = Join[-Rest[init] // Reverse, init]; lg = Length[init];
rep = {u[0] -> 314101616640, u[1] -> 25442230947840, u[2] -> 235226865664, u[3] -> -181502208, u[4] -> -16}; Clear[a];
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;
(* Print[Solve[rec == 0, a[n+4]][[1]] /. n -> n-4]; *)
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]));
Do[a[n] = init2[[n + lg]], {n, -(lg - 1), lg - 1}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 08 2018 *)

Crossrefs

Sequence in context: A260383 A056770 A061038 * A008894 A033973 A033356

Adjacent sequences: A058228 A058229 A058230 * A058232 A058233 A058234

Keyword

sign

Author

N. J. A. Sloane, Dec 02 2000

Status

approved