A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)*a(n-4) + 42*a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.

1, 1, 3, 13, 113, 1525, 57123, 2165017, 262621633, 42422452969, 14070212996451, 7658246457672229, 10650393355715621873, 15512114571284835412957, 75606222210863532170808003, 452005526897888844293504165425

Offset: 0

Author

Michael Somos, Jan 29 2023

Keywords

• nonn

Comments

The elliptic curve y^2 = x^3 - 2*x (LMFDB label 256.b1) has infinite order point P = (2, 2) and 2-torsion point T = (0, 0). The x and y coordinates of n*P + T have denominators a(n)^2 and a(n)^3 respectively.

Examples

			2*P + T = (-8/9, -28/27). 3*P + T  = (-1/169, 239/2197).
		

Links

• LMFDB, Elliptic Curve 256.b1 (Cremona label 256b1)

Crossrefs

Cf. A166929.

Programs

Mathematica

a[ m_] := With[{n = Abs[m]}, If[ n<3, {1, 1, 3}[[n+1]], (-a[n-1]*a[n-4] + 42*a[n-2]*a[n-3])/a[n-5]]];

PARI

{a(n) = my(E = ellinit([-2, 0])); sqrtint(denominator(elladd(E, [0, 0], ellmul(E, [2, 2], n))[1]))};

PARI

{a(n) = my(A); n = abs(n); A = vector(max(4, n+1), k, 1); A[3] = 3; A[4] = 13; for(k = 4, n, A[k+1] = (if(k%2, 4, 8)*A[k]*A[k-2] + A[k-1]^2)/A[k-3]); A[n+1]};

Formula

a(2*n-1) = A166929(n) for all n in Z.