A051138 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x and point (1, 0).

0, 1, 1, -1, -5, -4, 29, 129, -65, -3689, -16264, 113689, 2382785, 7001471, -398035821, -7911171596, 43244638645, 6480598259201, 124106986093951, -5987117709349201, -541051130050800400, -4830209396684261199

Offset: 0

Author

Michael Somos, Oct 12 1999

Keywords

• sign
• easy
• nice

Comments

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)).

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

The elliptic curve y^2 + y = x^3 - x has LMFDB label 37.a1 (Cremona label 37a1). - Michael Somos, Feb 07 2024

Examples

			G.f. = x + x^2 - x^3 - 5*x^4 - 4*x^5 + 29*x^6 + 129*x^7 - 65*x^8 + ...
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
• R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
• LMFDB, Elliptic Curve 37.a1
• Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
• Index to divisibility sequences

Crossrefs

Cf. A006769, A006720, A028937, A028939.

Programs

Mathematica

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 *)

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 ]

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 */

PARI

{a(n) = subst(elldivpol(ellinit([0, 0, 1, -1, 0]), n, 'x), 'x, 1)}; /* Michael Somos, Apr 21 2025 */

Formula

a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).

a(n) = (-a(n-1) * a(n-4) + 5 * a(n-2) * a(n-3)) / a(n-5).

a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3.

a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2.

a(-n) = -a(n). a(n) = A006769(2*n). a(n)^2 = A028937(n). |a(n)|^3 = A028939(n) for all n in Z.

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

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