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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.

%I A102276 #31 Jun 30 2024 22:08:40
%S A102276 1,1,1,1,1,1,2,3,4,8,17,50,107,239,1103,3775,14463,55283,256666,
%T A102276 2059753,9820288,55075036,503857819,4083736906,44590046729,
%U A102276 335845998321,3581731774609,68868876045617,782035904796497,11680434156713849,194342679446776442
%N 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.
%C A102276 Sequence defined by recursion derived from a genus 2 curve.
%C A102276 Similar to the Somos-6 and Somos-7 sequences with many bilinear identities.
%C A102276 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
%H A102276 Seiichi Manyama, <a href="/A102276/b102276.txt">Table of n, a(n) for n = 0..211</a>
%H A102276 A. J. van der Poorten, <a href="https://arxiv.org/abs/math/0412372">Curves of Genus 2, Continued Fractions and Somos Sequences</a>, arXiv:math/0412372 [math.NT], 2004.
%H A102276 A. J. van der Poorten, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Poorten2/vdp89.html">Curves of Genus 2, Continued Fractions and Somos Sequences</a>, J. Integer Seqs., 8 (2005), #05.3.4.
%H A102276 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F A102276 a(n) = A256858(2*n - 5) for all n in Z. - _Michael Somos_, Apr 13 2015
%F A102276 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
%F A102276 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
%F A102276 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
%t A102276 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 *)
%o A102276 (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])};
%o A102276 (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
%Y A102276 Cf. A006720, A006722, A256858, A256916, A018896, A271341, A271835, A271831, A271837, A271838, A271839.
%K A102276 nonn
%O A102276 0,7
%A A102276 _Michael Somos_, Jan 02 2005