A028937 -id:A028937 - cp's OEIS frontend

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

Michael Somos, Oct 12 1999

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

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

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

Cf. A006769, A006720, A028937, A028939.

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 *)

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 ]

{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 */

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

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

A028939 a(n) = denominator of y-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 1, 1, 125, 64, 24389, 2146689, 274625, 50202571769, 4302115807744, 1469451780501769, 13528653463047586625, 343216282443844010111, 63061816101171948456692661, 495133617181351428873673516736

Offset: 1

N. J. A. Sloane

			8P = (21/25, -69/125).

Seiichi Manyama, Table of n, a(n) for n = 1..86
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028936, A028937, A028938 (numerator), A028943.

P=(0, 0), 2P=(1, 0); if kP=(a, b) then (k+1)P = (a' = (b^2-a^3)/a^2, b' = -1 - b*a'/a).

a(n) = A028943(2n). - Seiichi Manyama, Nov 19 2016

A028936 Numerator of x-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 2, 6, 21, 161, 1357, 18526, 480106, 12551561, 683916417, 51678803961, 4881674119706, 997454379905326, 213822353304561757, 79799551268268089761, 53139223644814624290821, 36631192030206080565822006, 54202648602164057575419038802

Offset: 1

N. J. A. Sloane

			4P =(2, -3).
a(3) = 6 = 2*3 = A006720(4)*A006720(5). - _Michael Somos_, Apr 12 2020

Seiichi Manyama, Table of n, a(n) for n = 1..106
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028937 (denominator), A028938, A028939, A028940.

Cf. A006720.

a(n)=numerator(ellmul(E,[0,0],2*n)[1]) \\ Charles R Greathouse IV, Mar 23 2022

P=(0, 0), 2P=(1, 0); if kP = (a, b) then (k+1)P = (a' = (b^2 - a^3)/a^2, b' = -1 - b*a'/a).
a(n) = A028940(2n). - Seiichi Manyama, Nov 19 2016
0 = a(n)*a(n+6) - 5*a(n+1)*a(n+5) + 4*a(n+2)*a(n+4) - 20*a(n+3)^2 for all n in Z. a(n) = A006720(n+1)*A006720(n+2). - Michael Somos, Apr 12 2020

A028938 Negative of numerator of y-coordinate of (2n)*P where P is generator for rational points on curve y^2 + y = x^3 - x.

Original entry on oeis.org

0, 3, -14, 69, 2065, -28888, 2616119, -332513754, 8280062505, 18784454671297, -10663732503571536, 8938035295591025771, 31636113722016288336230, -41974401721854929811774227, 754388827236735824355996347601

Offset: 1

N. J. A. Sloane

			4P = (2, -3).

Seiichi Manyama, Table of n, a(n) for n = 1..86
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028936, A028937, A028939 (denominator), A028942.

P=(0, 0), 2P=(1, 0); if kP=(a, b) then (k+1)P = (a' = (b^2-a^3)/a^2, b' = -1 - b*a'/a).

a(n) = A028942(2n). - Seiichi Manyama, Nov 19 2016

cp's OEIS Frontend

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

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A028939 a(n) = denominator of y-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

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A028936 Numerator of x-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

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A028938 Negative of numerator of y-coordinate of (2n)*P where P is generator for rational points on curve y^2 + y = x^3 - x.

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