|
%I #37 Sep 08 2022 08:44:50
%S 1,1,1,1,8,27,343,12167,205379,30959144,3574558889,553185473329,
%T 578280195945297,238670664494938073,487424450554237378792,
%U 2035972062206737347698803,4801616835579099275862827431
%N a(n) = A006720(n)^3 (cubed terms of Somos-4 sequence).
%C 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.
%H Seiichi Manyama, <a href="/A028935/b028935.txt">Table of n, a(n) for n = 0..88</a>
%H B. Mazur, <a href="https://doi.org/10.1090/S0273-0979-1986-15430-3">Arithmetic on curves</a>, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.
%F 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).
%F 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
%e 5P = (1/4, -5/8).
%t 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 *)
%t 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 *)
%o (PARI) {b(n) = if(n< 4, 1, (b(n-1)*b(n-3) + b(n-2)^2)/b(n-4))};
%o for(n=0,30, print1((b(n))^3, ", ")) \\ _G. C. Greubel_, Feb 21 2018
%o (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
%Y Cf. A006720, A028934, A028943, A028945, A151502.
%K nonn
%O 0,5
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, May 14 2009
|
