A028940 -id:A028940 - cp's OEIS frontend

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.

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).

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
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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
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- see A028940.
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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).

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

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

N. J. A. Sloane, Jan 27 2022

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.

			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]

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

Cf. A028940-A028943, A350623-A350625.

\\ 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

N. J. A. Sloane, Jan 27 2022

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.

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

Cf. A028940-A028943, A350622-A350624.

PARI
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See A350622.
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A028944 Numerator of x-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, 1, -5, -20, 116, -3741, 8385, -239785, -59997896, 1849037896, -270896443865, -16683000076735, 2786836257692691, -3148929681285740316, 342115756927607927420, -280251129922563291422645, -804287518035141565236193151

Offset: 0

N. J. A. Sloane

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

Seiichi Manyama, Table of n, a(n) for n = 0..105
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028940, A028945.

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+1). - Seiichi Manyama, Nov 20 2016

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

N. J. A. Sloane, Jan 27 2022

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.

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

Cf. A028940-A028943, A350622-A350625.

PARI
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See A350622.
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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

N. J. A. Sloane, Jan 27 2022

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.

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

Cf. A028940-A028943, A350622-A350625.

PARI
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See A350622.
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A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).

Original entry on oeis.org

1, 2, 3, 14, 69, 413, 7222, 90211, 2577626, 127385577, 5092018073, 655664812074, 78618294607139, 13276948495989478, 5995083279193033837, 1895278734817024984181, 1542333923096758721461086, 1867485777936169465836858947, 2020248742951852823878208914098, 7078136335206254534330825538868049

Offset: 0

Thomas Scheuerle, Oct 22 2024

nonn

Consider the sequence s(k) with ordinary generating function: 1/(1-d(0)*x/(1-d(1)*x/(1-d(2)*x/(...)))), the Hankel sequence transform of s(k) is A006720 starting with the third term.
This is a special case of a more general theorem: Consider the sequence h(k) with generating function 1/(1-c(0)*x/(1-c(1)*x/(1-c(2)*x/(...)))), if c(2*k+1) = (c(2*k-2)*c(2*k-1) + t)/(c(2*k-4)*c(2*k-3)*c(2*k-2)^2*c(2*k-1)^2*c(2*k)) for all k with c(< 0) = 1, then the Hankel sequence transform of h(k) satisfies a Somos-4 A(1, t) recurrence.
If we would change the start condition into d(0..4) = {1,1,-1,-2,(5/2)}, the expansion of the continued fraction generating function would give us A171416, its Hankel sequence transform is again A006720. There exist infinitely many sequences with the same Hankel sequence transform.

Cf. A006720, A028940.

The Hankel transform is directly related to A006720: A157002, A157003, A160702, A171416, A173992, A173993, A254314.

d(n) = if(n<5, [1,1,1,2,1][n+1], (d(n-3)*d(n-2)+1)/(d(n-5)*d(n-4)*d(n-3)^2*d(n-2)^2*d(n-1)))
a(n) = numerator(d(2*n+1))

a(n) = -numerator(ellmul(ellinit([0, 3, -1, 2, 0]), [-1,0], 2*n+1)[1])

a(n) = A006720(n+1)*A006720(n+3).
denominator(d(2*n+1)) = A006720(n+2)^2.
-a(n)/A006720(n+3)^2 are the x-coordinates of (2*n+1) times [-1,0] on the curve y^2 - y = x^3 + 3*x^2 + 2*x. "Times" means here the multiplication under the elliptic group law.

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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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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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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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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.

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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.

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A028944 Numerator of x-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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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.

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A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).