A028945 - OEIS

%I #33 Apr 12 2020 20:53:19

%S 1,1,1,1,4,9,49,529,3481,98596,2337841,67387681,6941055969,

%T 384768368209,61935294530404,16063784753682169,2846153597907293521,

%U 2237394491744632911601,1262082793174195430038441,1063198259901027900600665796

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

%C 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.

%H Seiichi Manyama, <a href="/A028945/b028945.txt">Table of n, a(n) for n = 0..107</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) = (- 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

%F 5P = (1/4, -5/8).

%F 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

%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])^2, {n,0,30}] (* _G. C. Greubel_, Feb 21 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))^2, ", ")) \\ _G. C. Greubel_, Feb 21 2018

%o (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

%Y Cf. A006720, A028935, A028941, A028944.

%K nonn,frac

%O 0,5

%A _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_, May 14 2009