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A102276
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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.
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12
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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
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OFFSET
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0,7
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A006720, A006722, A256858, A256916, A018896, A271341, A271835, A271831, A271837, A271838, A271839.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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