OFFSET
0,4
COMMENTS
In Cantor (1994) on page 95 equation (1.8) equates psi(s) * psi(r) * psi(s+r) * psi(s-r) to the determinant of a 3 X 3 matrix in which each element is a product of psi(s+i) * psi(r+j) where i and j are between -2 and 2. - Michael Somos, Jul 18 2025
REFERENCES
D. G. Cantor (dgc(AT)ccrwest.org), email to N. J. A. Sloane, Nov. 30, 2000.
LINKS
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. See also New York J. Math, (2025) Vol. 31, 568-588. See pp. 5, 15-16.
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
0 = u4*a(n+5)*a(n-4) + u3*a(n+4)*a(n-3) + u2*a(n+3)*a(n-2) + u1*a(n+2)*a(n-1) + u0*a(n+1)*a(n) for all n in Z where u0=-15080550787549184, u1=-1722900809728, u2=39244793344, u3=1226959440, u4=8753. - Michael Somos, Jul 15 2025
0 = a(n)*( a(3)^3*a(n+2)*a(n-2) + a(2)^2*( a(5)*a(n+1)*a(n-1) - a(3)*a(n+3)*a(n-3) ) ) - a(2)*a(3)*a(4)*( a(n+1)^2*a(n-2) + a(n-1)^2*a(n+2) ) for all n in Z. - Michael Somos, Jul 18 2025
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 *)
Join[{0, 0}, RecurrenceTable[{a[n+4]*a[n-4] == 16*(-708993*a[n+3]*a[n-3] + 918854944*a[n+2]*a[n-2] + 99383714640*a[n+1]*a[n-1] + 1226959440*a[n]^2), a[2] == 1, a[3] == 36, a[4] == -16, a[5] == 5041728, a[6] == -19631351040, a[7] == -62024429150208, a[8] == -2805793044443561984, a[9] == -1213280369793911777918976}, a, {n, 2, 20}]] (* Michael Somos, May 23 2025 *)
PROG
(PARI) {a(n) = my(s = sign(n), v); if(n = abs(n), v = [0, 1, 36, -16, 5041728, -19631351040, -62024429150208, -2805793044443561984, -1213280369793911777918976]; if(n > 9, v = concat(v, vector(n-9)); for(k = 10, n, v[k] = 16*(-708993*v[k-1]*v[k-7] + 918854944*v[k-2]*v[k-6] + 99383714640*v[k-3]*v[k-5] + 1226959440*v[k-4]^2)/v[k-8])); s*v[n])}; /* Michael Somos, Jul 15 2025 */
(PARI) {a(n) = my(s = sign(n), v); if(n = abs(n), v = [0, 1, 36, -16, 5041728, -19631351040, -62024429150208]; if(n > 7, v = concat(v, vector(n-7)); for(k = 8, n, v[k] = (v[5]*v[k-4]*v[k-3]*v[k-2] - v[3]*v[4]*(v[k-5]*v[k-2]^2 + v[k-4]^2*v[k-1]) + v[3]^3*v[k-5]*v[k-3]*v[k-1]) / (v[3]*v[k-6]*v[k-3]))); s*v[n])}; /* Michael Somos, Jul 18 2025 */
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Dec 02 2000
STATUS
approved