A332828 Expansion of (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8) in powers of x.

0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 1, 0, 1, 1, 1, 1, 0, 1, -1, 1, -1, 0, -1, -1, -1, -1, 0, -1

Offset: 0

Author

Michael Somos, Feb 25 2020

Keywords

• sign
• easy

Comments

This is a (-1,1) generalized Somos-4 sequence.

For the elliptic curve y^2 + y = x^3 - x^2, the multiples of the point (0, 0) are (a(n-1)*a(n+1)/a(n)^2, -a(n-1)^2*a(n+2)/a(n)^3).

Examples

			G.f. = x + x^2 + x^3 + x^4 + x^6 - x^7 + x^8 - x^9 - x^11 - x^12 + ...
		

Links

• LMFDB, Elliptic Curve with LMFDB label 11.a3 (Cremona label 11a3)
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1,0,1,0,-1).

Crossrefs

Cf. A006720, A099443.

Programs

Mathematica

a[ n_] := {1, 1, 1, 1, 0, 1, -1, 1, -1, 0}[[Mod[n, 10, 1]]];

PARI

{a(n) = (-1)^(n\10) * [0, 1, 1, 1, 1, 0, 1, -1, 1, -1][n%10 + 1]};

PARI

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

Formula

G.f.: (x + x^2 + x^6 - x^7)/(1 - x^2 + x^4 - x^6 + x^8).

a(n) = -a(n+10) = a(5-n) for all n in Z.

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

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

a(2*n) = A099443(n-1), a(2*n+1) = A099443(n+2) for all n in Z.