A256858 -id:A256858 - cp's OEIS frontend

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

Michael Somos, Jan 02 2005

nonn

Sequence defined by recursion derived from a genus 2 curve.
Similar to the Somos-6 and Somos-7 sequences with many bilinear identities.
If a0 := a(n), a1 := a(n+1), ..., a5 := a(n+5), a6 := a(n+6) and a6 = (a5*a1 + a3^2)/a0 for all n in Z, then c := (a0^2*a1*a4*a5^2 + a0^2*a3*a4^3 + a1^3*a2*a5^2 + a0*a2^2*a3*a4^2 + a1^2*a2*a3^2*a5 + a0*a2*a3^3*a4 + a1*a2^3*a3*a5 + a2^3*a3^3)/(a0*a1*a2*a3*a4*a5) is constant. - Michael Somos, Jun 30 2024

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

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

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

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 *)

{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])};

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

A256916 a(n) = (a(n-1) * a(n-5) + a(n-3)^2) / a(n-6) with a(0) = a(1) = 1, a(2) = 0, a(3) = -1, a(4) = -3, a(8) = 29.

Original entry on oeis.org

1, 1, 0, -1, -3, -3, -2, 9, 29, 83, 56, -243, -2351, -7227, -18648, 54011, 698301, 5324929, 15128062, -28437275, -1438167267, -14356619593, -108319050672, 80689859625, 13472837856577, 268773209122329, 2678522836045616, 7565687047045511, -672545703786704803

Offset: 0

Michael Somos, Apr 13 2015

sign

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

G. C. Greubel, Table of n, a(n) for n = 0..208

Cf. A102276, A256858.

I:=[83, 56, -243, -2351, -7227, -18648]; [1,1,0,-1,-3,-3,-2,9,29] 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

Join[{1,1,0,-1,-3,-3,-2,9,29}, RecurrenceTable[{a[n] == (a[n-1]*a[n-5] + a[n-3]^2)/a[n-6], a[9] == 83, a[10] == 56, a[11] == -243, a[12] == -2351, a[13] == -7227, a[14] == -18648}, a, {n, 9, 60}]] (* G. C. Greubel, Aug 03 2018 *)

{a(n) = my(an); n = abs(n)+1; an = concat([ 1, 1, 0, -1, -3, -3, -2, 9, 29], vector(max(0, n-9), k)); for(k=10, n, an[k] = (an[k-1] * an[k-5] + an[k-3]^2) / an[k-6]); an[n]};

a(n) = a(-n) for all n in Z.
a(n) = A256858(2*n) for all n in Z.
Let b(n) = A102276(n). Then 0 = a(n) * b(n) - a(n+2) * b(n-2) + a(n+3) * b(n-3) for all n in Z.
0 = a(n) * a(n+6) - a(n+1) * a(n+5) - a(n+3) * a(n+3) for all n in Z.
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.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula