A377310 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 and point (1, 0).

0, 1, 1, 1, -3, -4, -13, 23, 87, 415, -152, -8063, -38727, -142471, 2309453, 13609844, 187790979, -1743980081, -25547499185, -575984295329, 1873521429456, 217675476797921, 5045023692031697, 65853623974941521, -5934036772012185603, -157454833217800083092

Offset: 0

Author

Michael Somos, Oct 23 2024

Keywords

• sign

Comments

Bisection of A210098 (even part). The other bisection is A277279.

The elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 has LMFDB label 57.a1 (Cremona label 57a1).

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = -3.

Examples

			G.f. = x + x^2 + x^3 - 3*x^4 - 4*x^5 - 13*x^6 + 23*x^7 + 87*x^8 + 415*x^9 + ...
		

Links

• C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
• LMFDB, Elliptic Curve 57a1

Crossrefs

Cf. A210098, A277279.

Programs

Maple

b:= proc(n) option remember; `if`(n<6, [0, 1$4, 2][n+1],
      (b(n-1)*b(n-4) -b(n-2)*b(n-3)) / b(n-5))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 05 2025

Mathematica

a[ n_] := a[n] = Which[ n<0, -a[-n], n<5, {0, 1, 1, 1, -3}[[n+1]], True, (a[n-1]*a[n-3] - a[n-2]^2)/a[n-4]];

PARI

{a(n) = my(v); if(n<0, -a(-n), n<5, [0, 1, 1, 1, -3][n+1], v = vector(n, i, if(i<5, a(i))); for(i=5, n, v[i] = (v[i-1]*v[i-3] - v[i-2]^2)/v[i-4]); v[n])};

PARI

{a(n) = my(E = ellinit([0, -1, 1, -2, 2]), z); z = ellpointtoz(E, [1, 0]); -(-1)^n*round(ellsigma(E, n*z)/ellsigma(E, z)^(n^2))};

PARI

{a(n) = my(E = ellinit([0, -1, 1, -2, 2])); sign(n) * subst( elldivpol( E, abs(n)), x, 1)};

Formula

a(n) = A210098(2*n).

a(n) = -a(-n) for all n in Z.

0 = a(n)*a(n+4) - a(n+1)*a(n+3) + a(n+2)^2 for all n in Z.

0 = a(n)*a(n+5) - a(n+1)*a(n+4) - 3*a(n+2)*a(n+3) for all n in Z.

0 = a(n+1)^2*a(n+2)^2 - a(n)^2*a(n+3)^2 - a(n)*a(n+2)^3 - a(n+1)^3*a(n+3) - 2*a(n)*a(n+1)*a(n+2)*a(n+3) for all n in Z.

Extensions

Duplicate term a(15)=2309453 removed by Georg Fischer, May 05 2025