A276402 A sequence related to the Somos-6 sequence A006722.

1, 1, -1, 1, -3, -3, 1, -25, 49, 1, 385, 1489, 503, 10753, -82371, -196419, -1575551, -12482641, 95770849, -739310591, 5684060161, 45018762529, 359479836623, 7751171129473, -59488778593731, -137028961472835, -29749898850946559, -441962130410844841

Offset: 0

Author

N. J. A. Sloane, Sep 12 2016

Keywords

• sign

Comments

This is the right-hand portion of a two-way infinite sequence ..., 1, -1, 1, 1, 1, 1, -1, 1, -3, -3, 1, -25, 49, 1, 385, 1489, 503, 10753, -82371, ...

0 = a(n+5)*a(n-4) + 2*a(n+4)*a(n-3) - 2*a(n+3)*a(n-2) - 4*a(n+2)*a(n-1) - 5*a(n+1)*a(n) = 2*a(n+5)*a(n-5) + 4*a(n+4)*a(n-4) + 5*a(n+3)*a(n-3) + 19*a(n+2)*a(n-2) + 18*a(n+1)*a(n-1) for all n in Z. - Michael Somos, Aug 14 2023

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..206
• Yuri N. Fedorov and Andrew N. W. Hone, Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties, arXiv:1512.00056 [nlin.SI], 2015-2016. See Eq. (6.8).

Crossrefs

Cf. A006722.

Programs

Mathematica

a[0] = a[1] = 1; a[2] = -1; a[3] = 1; a[4] = a[5] = -3;
a[n_] := a[n] = (-2a[n-3]^2 + 2a[n-4] a[n-2] + a[n-5] a[n-1])/a[n-6];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 13 2018, after Seiichi Manyama *)

PARI

{a(n) = if( n>-1 && n<3, [1, 1, -1][n+1], if( n<0, a(-1-n), (a(n-1)*a(n-5) + 2*a(n-2)*a(n-4) - 2*a(n-3)*a(n-3)) / a(n-6)))}; /* Michael Somos, Aug 14 2023 */

Formula

a(0) = a(1) = 1, a(2) = -1, a(3) = 1, a(4) = a(5)= -3, a(n+6)*a(n) = a(n+5)*a(n+1) + 2*a(n+4)*a(n+2) - 2*a(n+3)^2. - Seiichi Manyama, Sep 12 2016

a(n) = a(-1-n) for all n in Z. - Michael Somos, Aug 14 2023

Extensions

More terms from Seiichi Manyama, Sep 12 2016