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%I #43 Feb 08 2024 01:44:37
%S 0,1,1,-1,-5,-4,29,129,-65,-3689,-16264,113689,2382785,7001471,
%T -398035821,-7911171596,43244638645,6480598259201,124106986093951,
%U -5987117709349201,-541051130050800400,-4830209396684261199
%N Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x and point (1, 0).
%C This is a strong divisibility sequence; that is, if n divides m, then a(n) divides a(m) and moreover for all positive integer n,m a(gcd(n, m)) = gcd(a(n), a(m)).
%C This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = -5. - _Michael Somos_, Jul 07 2014
%C The elliptic curve y^2 + y = x^3 - x has LMFDB label 37.a1 (Cremona label 37a1). - _Michael Somos_, Feb 07 2024
%H T. D. Noe, <a href="/A051138/b051138.txt">Table of n, a(n) for n = 0..100</a>
%H Clark Kimberling, <a href="http://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H R. W. Gosper and Richard C. Schroeppel, <a href="http://arxiv.org/abs/math/0703470">Somos Sequence Near-Addition Formulas and Modular Theta Functions</a>, arXiv:math/0703470 [math.NT], 2007.
%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/37/a/1">Elliptic Curve 37.a1</a>
%H Helmut Ruhland, <a href="https://arxiv.org/abs/2312.02085">Somos-4 and a quartic Surface in RP^3</a>, arXiv:2312.02085 [math.AG], 2023.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).
%F a(n) = (-a(n-1) * a(n-4) + 5 * a(n-2) * a(n-3)) / a(n-5).
%F a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3.
%F a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2.
%F a(-n) = -a(n). a(n) = A006769(2*n). a(n)^2 = A028937(n). |a(n)|^3 = A028939(n) for all n in Z.
%F 0 = a(n)*a(n+4) - a(n+1)*a(n+3) - a(n+2)*a(n+2) for all n in Z. - _Michael Somos_, Jul 07 2014
%F 0 = a(n)*a(n+5) + a(n+1)*a(n+4) - 5*a(n+2)*a(n+3) for all n in Z. - _Michael Somos_, Jul 07 2014
%e G.f. = x + x^2 - x^3 - 5*x^4 - 4*x^5 + 29*x^6 + 129*x^7 - 65*x^8 + ...
%t a[n_ /; n < 0] := -a[-n]; a[0] = 0; A006769[n_] := (ClearAll[an]; an[_] = 1; an[3] = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]); a[n_] := A006769[2n]; Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Apr 11 2012, after 2nd formula *)
%o (PARI) an = vector(200); an = concat([ 1, 1, -1, -5 ], an); for( n=5, length(an), an[ n ]=(an[ n-1 ] * an[ n-3 ] + an[ n-2 ]^2) / an[ n-4 ]); a(n) = an[ n ]
%o (PARI) {a(n) = my(v = [1, 1, -1, -5]); if( n<0, -a(-n), if( n==0, 0, if( n<5, v[n], v = concat( v, vector(n - 4)); for( k=5, n, v[k] = (v[k-1] * v[k-3] + v[k-2]^2) / v[k-4]); v[n])))}; /* _Michael Somos_, Feb 12 2012 */
%Y Cf. A006769, A006720, A028937, A028939.
%K sign,easy,nice
%O 0,5
%A _Michael Somos_, Oct 12 1999
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