cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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

References

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

Links

Crossrefs

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

Programs

  • PARI
    \\ 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

Formula

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

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

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

References

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

Links

Crossrefs

Cf. A028940, A028941, A028943, A278314.

Programs

Formula

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).

A028941 Denominator of X-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, 4, 1, 9, 25, 49, 16, 529, 841, 3481, 16641, 98596, 4225, 2337841, 13608721, 67387681, 264517696, 6941055969, 12925188721, 384768368209, 5677664356225, 61935294530404, 49020596163841, 16063784753682169
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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 denominators of the x_n. - N. J. A. Sloane, Jan 27 2022
Squares of terms in A006769 (or A006720).

References

  • G. Everest, A. J. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences: Examples and Applications, AMS Monographs, 2003
  • A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Links

Crossrefs

Cf. A028940, A028942, A028943.

Programs

  • Mathematica
    nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,((c*2e)-f*b+2d^2)/a}; NestList[nxt,{1,1,1,1,4,1},30][[;;,1]] (* Harvey P. Dale, Dec 26 2024 *)
  • PARI
    - see A028940.

Formula

This sequence satisfies the quadratic recurrence relation a(n)a(n-6)=-a(n-1)a(n-5)+2a(n-2)a(n-4)+2a(n-3)^2 which is a generalized Somos-6 relation. - Graham Everest (g.everest(AT)uea.ac.uk), Dec 16 2002
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).

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

Views

Author

N. J. A. Sloane

Keywords

Examples

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

Links

Crossrefs

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

Formula

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

A028935 a(n) = A006720(n)^3 (cubed terms of Somos-4 sequence).

Original entry on oeis.org

1, 1, 1, 1, 8, 27, 343, 12167, 205379, 30959144, 3574558889, 553185473329, 578280195945297, 238670664494938073, 487424450554237378792, 2035972062206737347698803, 4801616835579099275862827431
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A006720, A028934, A028943, A028945, A151502.

Programs

  • Magma
    I:=[1,1,1,1,8,27,343,12167,205379]; [n le 9 select I[n] else (129*Self(n-1)*Self(n-8) - 260*Self(n-2)*Self(n-7) - 8385*Self(n-3)*Self(n-6) + 48633*Self(n-4)*Self(n-5))/Self(n-9): n in [1..30]]; // G. C. Greubel, Feb 22 2018
  • Mathematica
    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])^3, {n,0,30}] (* G. C. Greubel, Feb 21 2018 *)
a[n_ /; 0 <= n <= 3] = 1; a[4]:= 8; a[5]:= 27; a[6]:= 343; a[7]:= 12167; a[8]:= 205379; a[9]:= 30959144; a[n_]:= a[n] = (129*a[n-1]*a[n-8] - 260*a[n-2]*a[n-7] - 8385*a[n-3]*a[n-6] + 48633*a[n-4]*a[n-5])/a[n-9]; Table[a[n], {n,0,30}] (* G. C. Greubel, Feb 22 2018 *)
  • PARI
    {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))^3, ", ")) \\ G. C. Greubel, Feb 21 2018

Formula

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) = (129*a(n-1)*a(n-8) - 260*a(n-2)*a(n-7) - 8385*a(n-3)*a(n-6) + 48633*a(n-4)*a(n-5))/a(n-9). - G. C. Greubel, Feb 22 2018

Extensions

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

A350622 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^2.

Original entry on oeis.org

0, -1, 1, 2, -3, -2, 21, 11, -140, 209, 1740, -3629, -17139, 194438, 528157, -8338438, 15659721, 665838199, -1524968280, -50443970239, 791991662680, 8985658531079, -211327567932999, 38581695555082, 112336114570262877, -20672037869235082, -59711116955990028899, 1404304980033091755971, 63734020523767895773980, -2251247528715575677121711, -33058398375463796474831580
Offset: 1

Views

Author

N. J. A. Sloane, Jan 27 2022

Keywords

Comments

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.

Examples

			P[1] to P[16] are [0, 0], [-1, -1], [1, -2], [2, 3], [-3/4, 1/8], [-2/9, -28/27], [21, -99], [11/49, 20/343], [-140/121, -931/1331], [209/400, -10527/8000], [1740/361, 76400/6859], [-3629/7569, 71117/658503], [-17139/36481, -7705242/6967871], [194438/38809, -97805561/7645373], [528157/1036324, 317884519/1054977832], [-8338438/7187761, -6053168484/19270387241]

References

  • D. Husemoller, Elliptic Curves, Springer, 1987, p. 28.
  • A. W. Knapp, Elliptic Curves, Princeton, 1992, p. 64.

Crossrefs

Cf. A028940-A028943, A350623-A350625.

Programs

  • PARI
    \\ To get the first 40 points P[n].
\\ define curve E
E = ellinit([0,1,1,0,0])
P[1] = [0,0]
for(n=2, 40, P[n] = elladd(E, P[1], P[n-1]))
P

A350625 a(n) = denominator of the Y-coordinate of n*P where P is the generator [0,0] for rational points on the curve y^2 + y = x^3 + x^2.

Original entry on oeis.org

1, 1, 1, 1, 8, 27, 1, 343, 1331, 8000, 6859, 658503, 6967871, 7645373, 1054977832, 19270387241, 549554511871, 199279038321, 537149706740569, 17795935051712000, 238963978065144151, 27915217583090079761, 3036108535167687186689, 338086202776927409397159
Offset: 1

Views

Author

N. J. A. Sloane, Jan 27 2022

Keywords

Comments

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 denominators of the y_n.

References

  • D. Husemoller, Elliptic Curves, Springer, 1987, p. 28.
  • A. W. Knapp, Elliptic Curves, Princeton, 1992, p. 64.

Crossrefs

Cf. A028940-A028943, A350622-A350624.

Programs

A278314 a(n) = -c(n-1) * c(n-2) * c(n+3) where c(n) = A006769(n).

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, -588310630753491921045
Offset: 1

Views

Author

Michael Somos, Nov 17 2016

Keywords

Comments

a(n) = A028942(n) up to sign.
y coordinate of n*P = -A028942(n) / A028943(n) = a(n) / A006769(n)^3 where P is generator for rational points on curve y^2 + y = x^3 - x.

Examples

			G.f. = x^3 - 3*x^4 - 5*x^5 - 14*x^6 - 8*x^7 + 69*x^8 - 435*x^9 + ...

Links

Crossrefs

Cf. A006769, A028942, A028943.

Programs

  • PARI
    {a(n) = my(m, an); if( n>0, m = n; an = vector( max(12, m), i, if( i<13, [0, 0, 1, -3, -5, -14, -8, 69, -435, 2065, 3612, 28888][i], 0)), m = 1-n; an = vector( max(12, m), i, if( i<13, [1, 1, 1, 0, -2, 3, -15, -35, -56, -92, 2001, -8555][i], 0))); for( k=13, m, an[k] = (an[k-1] * an[k-7] + 3 *  an[k-2] * an[k-6] - 3 * an[k-3] * an[k-5] + 6 * an[k-4]^2) / an[k-8]); an[m]};

Formula

0 = a(n)*a(n+8) - a(n+1)*a(n+7) - 3*a(n+2)*a(n+6) + 3*a(n+3)*a(n+5) - 6*a(n+4)^2 for all n in Z.
0 = a(n+1)*a(n+2)*a(n+6) - 2*a(n+1)*a(n+3)*a(n+5) + 3*a(n+1)*a(n+4)^2 + 3*a(n+2)^2*a(n+5) + a(n+2)*a(n+3)*a(n+4) - a(n+3)^3 for all n in Z.

A350623 a(n) = denominator 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^2.

Original entry on oeis.org

1, 1, 1, 1, 4, 9, 1, 49, 121, 400, 361, 7569, 36481, 38809, 1036324, 7187761, 67092481, 34117281, 6607901521, 68162766400, 385083543601, 9202249657441, 209674135856641, 4853089476046161, 7099336433764, 2600282294202480889, 60193393235277536641, 1371165544633857017809
Offset: 1

Views

Author

N. J. A. Sloane, Jan 27 2022

Keywords

Comments

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 denominators of the x_n.

References

  • D. Husemoller, Elliptic Curves, Springer, 1987, p. 28.
  • A. W. Knapp, Elliptic Curves, Princeton, 1992, p. 64.

Crossrefs

Cf. A028940-A028943, A350622-A350625.

Programs

A350624 a(n) = numerator of the Y-coordinate of n*P where P is the generator [0,0] for rational points on the curve y^2 + y = x^3 + x^2.

Original entry on oeis.org

0, -1, -2, 3, 1, -28, -99, 20, -931, -10527, 76400, 71117, -7705242, -97805561, 317884519, -6053168484, -584285903929, 17516504939480, 21171512841831, -20045208029885441, -987005650468865600, 26826505806361752519, -24519007717765931978, -338107738763085297600203, 37652404140584119758794769, 262883121764561512399492, -470660250581978416129759599211, -103603683448954712692908522816060, 17053994466435658069907361489699701
Offset: 1

Views

Author

N. J. A. Sloane, Jan 27 2022

Keywords

Comments

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 y_n.

References

  • D. Husemoller, Elliptic Curves, Springer, 1987, p. 28.
  • A. W. Knapp, Elliptic Curves, Princeton, 1992, p. 64.

Crossrefs

Cf. A028940-A028943, A350622-A350625.

Programs

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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