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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

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.

Links

• 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.

Crossrefs

Cf. A006720, A028935, A028941, A028944.

Programs

Magma

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

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])^2, {n,0,30}] (* G. C. Greubel, Feb 21 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))^2, ", ")) \\ 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) = (- 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

Extensions

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