A129206 First sequence in solution to congruent number 5 problem.

0, 1, -12, -2257, 1494696, 8914433905, -178761481355556, -62419747600438859233, -5354229862821602092291248, 1001926359199672697329083442936609, 50016678000996026579336936742637753055940

Offset

0,3

Comments

Let W(n)=A129206(n), X(n)=A129207(n), Y(n)=A129208(n), Z(n)=A129209(n).

These four sequences correspond to the four Jacobi theta functions or Weierstrass sigma functions.

References

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, 1939. See p. 427.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..49

Michael Somos, Weierstrass Elliptic Function Polynomials (PARI/GP .gp version) or (Mathematica .wl version).

Formula

Right triangle with sides |10*Y(n)*W(n) / (X(n)*Z(n))|, |X(n)*Z(n) / (Y(n)*W(n))|, |2*Y(2*n) / W(2*n)| has area 5.

W(2*n) = (-1)^n * 2 * W(n) * X(n) * Y(n) * Z(n) for all n in Z.

a(n+2) * a(n-2) = 144 * a(n+1) * a(n-1) + 2257 * a(n)^2, a(n) = -a(-n) for all n in Z.

Prog

(PARI) {a(n) = my(s=sign(n)); n=abs(n); if( n<1, 0, s * if( n<5, [1, -12, -2257, 1494696][n], (144 * a(n-1) * a(n-3) + 2257 * a(n-2)^2 ) / a(n-4) ))};

Crossrefs

Cf. A129207, A129208, A129209.

Sequence in context: A268071 A205306 A268421 * A135398 A203426 A268589

Adjacent sequences: A129203 A129204 A129205 * A129207 A129208 A129209

Keyword

sign

Author

Michael Somos, Apr 03 2007

Status

approved