A028942 -id:A028942 - cp's OEIS frontend

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.

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
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See A028940.
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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

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 denominators of the x_n. - N. J. A. Sloane, Jan 27 2022

Squares of terms in A006769 (or A006720).

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.

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. A028940, A028942, A028943.

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.
```

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).

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

A028934 Negative of numerator of y-coordinate of (2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

0, 1, 5, -8, 435, 3612, 43355, 28076979, -331948240, 641260644409, 318128427505160, -66316334575107447, 588310630753491921045, 435912379274109872312968, 2181616293371330311419201915

Offset: 0

N. J. A. Sloane

			3P = (-1, -1). 5P = (1/4, -5/8). 7P = (-5/9, 8/27).

Seiichi Manyama, Table of n, a(n) for n = 0..86
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028935, A028942.

a[ n_] := If[n == 0, 0, -Numerator[ #[[3]]/#[[1]]^3 & @ Nest[Function[z, Module[{w, x, y}, {w, x, y} = z; {w x, y^2 - x^3, -y (y^2 - x^3) - (w x)^3}]], {1, 1, 0}, 2 n - 1]]]; (* Michael Somos, Apr 13 2020 *)

{a(n) = -numerator(ellmul(ellinit([0, 0, 1, -1, 0]), [0, 0], 2*n+1)[2])}; /* Michael Somos, Apr 13 2020 */

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+1). - Seiichi Manyama, Nov 20 2016
0 = a(n)*a(n+8) -145*a(n+1)*a(n+7) +3225*a(n+2)*a(n+6) -18705*a(n+3)*a(n+5) +14964*a(n+4)*a(n+4) for all n in Z. - Michael Somos, Apr 13 2020

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

Michael Somos, Nov 17 2016

sign

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.

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

Seiichi Manyama, Table of n, a(n) for n = 1..173

Cf. A006769, A028942, A028943.

{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]};

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.

A334068 Negative of numerator of y-coordinate of -(2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 0, 3, 35, -92, 8555, 162024, 2882165, 3906507129, -88075171080, 260151768440137, 304986999070045520, -100886180199254542253, 1600059682932627475385835, 2620000542207768964443625516

Offset: 0

Michael Somos, Apr 13 2020

sign

			-P = (0, -1), -3P = (-1 ,0), -5P = (1/4, -3/8), -7P = (-5/9, -35/27).

Robert Israel, Table of n, a(n) for n = 0..86

Cf. A028934, A028942.

f:= proc(m) option remember; -(-145*procname(m - 7)*procname(m - 1) + 3225*procname(m - 6)*procname(m - 2) - 18705*procname(m - 5)*procname(m - 3) + 14964*procname(m - 4)^2)/procname(m - 8) end proc:
Data:= [1, 0, 3, 35, -92, 8555, 162024, 2882165, 3906507129, -88075171080]:
for i from 0 to 9 do f(i):= Data[i+1] od:
map(f, [$0..20]); # Robert Israel, Oct 06 2020

{a(n) = -numerator(ellmul(ellinit([0, 0, 1, -1, 0]), [0, 0], -2*n-1)[2])};

a(n) = A028934(-1-n) = A028942(-2*n-1) for all n in Z.

0 = a(n)*a(n+8) -145*a(n+1)*a(n+7) +3225*a(n+2)*a(n+6) -18705*a(n+3)*a(n+5) +14964*a(n+4)*a(n+4) for all n in Z.

cp's OEIS Frontend

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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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.

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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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A028934 Negative of numerator of y-coordinate of (2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

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Mathematica

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

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A334068 Negative of numerator of y-coordinate of -(2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

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