A102276 a(n) = (a(n-1) * a(n-5) + a(n-3)^2) / a(n-6) with a(0) = ... = a(5) = 1, a(n) = a(5-n) for all n in Z.

1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, 107, 239, 1103, 3775, 14463, 55283, 256666, 2059753, 9820288, 55075036, 503857819, 4083736906, 44590046729, 335845998321, 3581731774609, 68868876045617, 782035904796497, 11680434156713849, 194342679446776442

Offset

0,7

Comments

Sequence defined by recursion derived from a genus 2 curve.

Similar to the Somos-6 and Somos-7 sequences with many bilinear identities.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..211

A. J. van der Poorten, Curves of Genus 2, Continued Fractions and Somos Sequences, arXiv:math/0412372 [math.NT], 2004.

A. J. van der Poorten, Curves of Genus 2, Continued Fractions and Somos Sequences, J. Integer Seqs., 8 (2005), #05.3.4.

Index entries for two-way infinite sequences

Formula

a(n) = A256858(2*n - 5) for all n in Z. - Michael Somos, Apr 13 2015

Let b(n) = A256916(n). Then 0 = a(n) * b(n) - a(n-2) * b(n+2) + a(n-3) * b(n+3) for all n in Z. - Michael Somos, Apr 13 2015

0 = a(n) * a(n+6) - a(n+1) * a(n+5) - a(n+3) * a(n+3) for all n in Z. - Michael Somos, Apr 13 2015

0 = a(n) * a(n+9) + a(n+2) * a(n+7) - a(n+3) * a(n+6) - 9 * a(n+4) * a(n+5) for all n in Z. - Michael Somos, Apr 13 2015

Mathematica

Join[{1, 1, 1, 1, 1}, RecurrenceTable[{a[n] == (a[n-1]*a[n-5] + a[n-3]^2)/a[n-6], a[6] == 1, a[7] == 2, a[8] == 3, a[9] == 4, a[10] == 8, a[11] == 17}, a, {n, 6, 60}]] (* G. C. Greubel, Aug 03 2018 *)

Prog

(PARI) {a(n) = my(an); if( n<0, a(5-n), n++; an = vector(n, i, 1); for(k=7, n, an[k] = (an[k-1]*an[k-5] + an[k-3]^2) / an[k-6]); an[n])};
(Magma) I:=[1, 2, 3, 4, 8, 17]; [1, 1, 1, 1, 1] cat [n le 6 select I[n] else (Self(n-1)*Self(n-5) + Self(n-3)^2)/Self(n-6): n in [1..30]]; // G. C. Greubel, Aug 03 2018

Crossrefs

Cf. A006720, A006722, A256858, A256916, A018896, A271341, A271835, A271831, A271837, A271838, A271839.

Sequence in context: A349763 A360000 A296109 * A215897 A276673 A282815

Adjacent sequences: A102273 A102274 A102275 * A102277 A102278 A102279

Keyword

nonn

Author

Michael Somos, Jan 02 2005

Status

approved