A028941 -id:A028941 - cp's OEIS frontend

A006769 Elliptic divisibility sequence associated with elliptic curve "37a1": y^2 + y = x^3 - x and multiples of the point (0,0).

0, 1, 1, -1, 1, 2, -1, -3, -5, 7, -4, -23, 29, 59, 129, -314, -65, 1529, -3689, -8209, -16264, 83313, 113689, -620297, 2382785, 7869898, 7001471, -126742987, -398035821, 1687054711, -7911171596, -47301104551, 43244638645

Offset: 0

Michael Somos, Jul 16 1999

This sequence has a recursion same as the Somos-4 sequence recursion.
a(n+1) is the Hankel transform of A178072. - Paul Barry, May 19 2010
The recurrence formulas in [Kimberling, p. 16] are missing square and cube exponents. - Michael Somos, Jul 07 2014
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = 1.
From Helmut Ruhland, Nov 28 2023: (Start)
This sequence and its two subsequences with even/odd indices satisfy the Somos-4 recursion.
The even subsequence is A051138, here called r[ ]. The odd subsequence is the classical Somos-4 A006720, here called s[ ].
These two subsequences interleaved as follows, recover the original sequence which is now: r[0], s[2], r[1], -s[3], r[2], s[4], r[3], -s[5], ...,  all Somos-4 s[ ] with odd index with a minus sign. (End)

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 11 and 164.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..300 (first 101 terms from T. D. Noe)
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulas for hyperelliptic sigma functions, arXiv:math/0501162 [math.NT], 2005.
Graham Everest, Elliptic Divisibility Sequences.
R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
LMFDB, Elliptic Curve 37.a1 (Cremona label 37a1)
Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
Michael Somos, Number Walls in Combinatorics
Index entries for two-way infinite sequences

Cf. A006720, A028941, A050512, A051138, A178072, A278314.

a006769 n = a050512_list !! n
a006769_list = 0 : 1 : 1 : (-1) : 1 : zipWith div (zipWith (+) (zipWith (*)
   (drop 4 a006769_list) (drop 2 a006769_list))
     (map (^ 2) (drop 3 a006769_list))) (tail a006769_list)
-- Reinhard Zumkeller, Nov 02 2011

a[n_] := If[n < 0, -a[-n], If[n == 0, 0, 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]]]; Table[a[n], {n, 0, 32}] (* _Jean-François Alcover, Dec 14 2011, after first Pari program *)
Join[{0},RecurrenceTable[{a[1]==a[2]==1,a[3]==-1,a[4]==1,a[n]==(a[n-1] a[n-3]+ a[n-2]^2)/a[n-4]},a,{n,40}]] (* Harvey P. Dale, May 04 2018 *)
a[ n_] := Which[n<0, -a[-n], n<5, {0, 1, 1, -1, 1}[[1+n]], True, (a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]]; (* Michael Somos, Aug 20 2024 *)

{a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(3, n), i, 1); an[3] = -1; for( k=5, n, an[k] = (an[k-1] * an[k-3] + an[k-2]^2) / an[k-4]); an[n]))};

{a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = Vec((-1 - 2*x + sqrt(1 + 4*x - 4*x^3 + O(x^n))) / (2 * x^2)); matdet( matrix((n-1)\2, (n-1)\2, i, j, if(i + j - 1 - n%2<0, 0, an[i + j -n%2])))))};

{a(n) = my(E, z); E = ellinit([0, 0, -1, -1, 0]); z = ellpointtoz(E, [0, 0]); round( ellsigma(E, n*z) / ellsigma(E, z)^(n^2))}; /* Michael Somos, Oct 22 2004 */

{a(n) = sign(n) * subst( elldivpol( ellinit([0, 0, -1, -1, 0]), abs(n)), x, 0)}; /* Michael Somos, Dec 16 2014 */

a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4) for all n != 4.
a(n) = (-a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5) for all n != 5.
a(-n) = -a(n) for all n.
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 for all n.
A006720(n) = (-1)^n * a(2*n - 3), A028941(n) = a(n)^2 for all n.
a(2*n) = A051138(n). - Michael Somos, Feb 10 2015
a(2*n+1) = a(n-1)*a(n)^2*a(n+3) - a(n-2)*a(n+1)^2*a(n+2) for all n. - Michael Somos, Aug 20 2024

A028940 a(n) = numerator of the X-coordinate of n*P where P is the generator [0,0] for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

0, 1, -1, 2, 1, 6, -5, 21, -20, 161, 116, 1357, -3741, 18526, 8385, 480106, -239785, 12551561, -59997896, 683916417, 1849037896, 51678803961, -270896443865, 4881674119706, -16683000076735, 997454379905326

Offset: 1

N. J. A. Sloane

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives numerators of the x_n. - N. J. A. Sloane, Jan 27 2022

			4P = P[4] = [2, -3].
P[1] to P[16] are [0, 0], [1, 0], [-1, -1], [2, -3], [1/4, -5/8], [6, 14], [-5/9, 8/27], [21/25, -69/125], [-20/49, -435/343], [161/16, -2065/64], [116/529, -3612/12167], [1357/841, 28888/24389], [-3741/3481, -43355/205379], [18526/16641, -2616119/2146689], [8385/98596, -28076979/30959144], [480106/4225, 332513754/274625]. - _N. J. A. Sloane_, Jan 27 2022

A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Seiichi Manyama, Table of n, a(n) for n = 1..212
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006769, A028941, A028942, A028943, A350622-A350625.

\\ from N. J. A. Sloane, Jan 27 2022. To get the first 40 points P[n].
\\ define curve E
E = ellinit([0,0,1,-1,0]) \\ y^2+y = x^3-x
P = vector(100)
P[1] = [0,0]
for(n=2, 40, P[n] = elladd(E, P[1], P[n-1]))
P

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) = a(n) = - A006769(n-1) * A006769(n+1) for all n in Z. - Michael Somos, Jul 28 2016

A028943 Denominator of y coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 27, 125, 343, 64, 12167, 24389, 205379, 2146689, 30959144, 274625, 3574558889, 50202571769, 553185473329, 4302115807744, 578280195945297, 1469451780501769, 238670664494938073, 13528653463047586625

Offset: 1

N. J. A. Sloane

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives numerators of the x_n. - N. J. A. Sloane, Jan 27 2022

			5P = (1/4, -5/8).

A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Seiichi Manyama, Table of n, a(n) for n = 1..173
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028940, A028941, A028942.

PARI
```
See A028940.
```

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).

A028942 Negative of numerator of y coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

Original entry on oeis.org

0, 0, 1, 3, 5, -14, -8, 69, 435, 2065, 3612, -28888, 43355, 2616119, 28076979, -332513754, -331948240, 8280062505, 641260644409, 18784454671297, 318128427505160, -10663732503571536, -66316334575107447, 8938035295591025771

Offset: 1

N. J. A. Sloane

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives negated numerators of the y_n. - N. J. A. Sloane, Jan 27 2022

a(n) = A278314(n) up to sign. - Michael Somos, Nov 19 2016

			3P = (-1, -1),
4P = (2, -3),
5P = (1/4, -5/8),
6P = (6, 14).

A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Seiichi Manyama, Table of n, a(n) for n = 1..173
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028940, A028941, A028943, A278314.

PARI
```
- see A028940.
```

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).

A028937 Denominator of x-coordinate of (2n)*P where P = (0,0) is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 1, 1, 25, 16, 841, 16641, 4225, 13608721, 264517696, 12925188721, 5677664356225, 49020596163841, 158432514799144041, 62586636021357187216, 1870098771536627436025, 41998153797159031581158401, 15402543997324146892198790401

Offset: 1

N. J. A. Sloane

			a(4) = 25 where 8P = (21/25, -69/125).

Seiichi Manyama, Table of n, a(n) for n = 1..106
LMFDB, Elliptic Curve 37.a1 (Cremona label 37a1)
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006720, A051138, A028936 (numerator), A028938, A028939, A028941.

a(n)=denominator(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) = A028941(2n). - Seiichi Manyama, Nov 19 2016
a(n) = a(-n) = b(n)*b(n+3) - b(n+1)*b(n+2) for all n in Z where b(n) = A006720(n). - Michael Somos, Mar 23 2022

A028945 a(n) = A006720(n)^2 (squared terms of Somos-4 sequence).

Original entry on oeis.org

1, 1, 1, 1, 4, 9, 49, 529, 3481, 98596, 2337841, 67387681, 6941055969, 384768368209, 61935294530404, 16063784753682169, 2846153597907293521, 2237394491744632911601, 1262082793174195430038441, 1063198259901027900600665796

Offset: 0

N. J. A. Sloane

If first two 1's are omitted, denominator of x-coordinate of (2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

Seiichi Manyama, Table of n, a(n) for n = 0..107
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006720, A028935, A028941, A028944.

I:=[1,1,1,1,4,9,49]; [n le 7 select I[n] else (- 4*Self(n-6)*Self(n-1) + 29*Self(n-5)*Self(n-2) + 116*Self(n-4)*Self(n-3) )/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

b[n_ /; 0 <= n <= 4] = 1; b[n_]:= b[n] = (b[n-1]*b[n-3] + b[n-2]^2)/b[n -4]; Table[(b[n])^2, {n,0,30}] (* G. C. Greubel, Feb 21 2018 *)

{b(n) = if(n< 4, 1, (b(n-1)*b(n-3) + b(n-2)^2)/b(n-4))};
for(n=0,30, print1((b(n))^2, ", ")) \\ G. C. Greubel, Feb 21 2018

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) = (- 4 a(n - 6) a(n - 1) + 29 a(n - 5) a(n - 2) + 116 a(n - 4) a(n - 3))/a(n-7). - Bill Gosper, May 14 2009
5P = (1/4, -5/8).
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. - Michael Somos, Apr 12 2020

Edited by N. J. A. Sloane, May 14 2009

cp's OEIS Frontend

A006769 Elliptic divisibility sequence associated with elliptic curve "37a1": y^2 + y = x^3 - x and multiples of the point (0,0).

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A028940 a(n) = numerator of the X-coordinate of n*P where P is the generator [0,0] for rational points on the curve y^2 + y = x^3 - x.

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A028943 Denominator of y coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

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A028942 Negative of numerator of y coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

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A028937 Denominator of x-coordinate of (2n)*P where P = (0,0) is the generator for rational points on the curve y^2 + y = x^3 - x.

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A028945 a(n) = A006720(n)^2 (squared terms of Somos-4 sequence).

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