OFFSET
0,5
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.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..88
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.
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
EXAMPLE
5P = (1/4, -5/8).
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, May 14 2009
STATUS
approved