A129209 Fourth sequence in solution to congruent number 5 problem.

1, 3, 31, 5283, -113279, 21166249443, -518493692732129, 189797666150873887683, -249563579992463717493803519, 2960896329211804542556804051252803, -115038188620995226180802686473825513089249

Offset

0,2

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

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.

Z(2*n) = Z(n)^4 -50* W(n)^4.

a(n+2) * a(n-2) = -144*a(n+1) * a(n-1) +2257 * a(n)^2. a(-n) = a(n).

Prog

(PARI) {a(n) = n=abs(n); if( n<1, 1, if(n<4, [3, 31, 5283][n], (-144 * a(n-1) * a(n-3) + 2257* a(n-2)^2 ) / a(n-4) ))};

Crossrefs

Sequence in context: A201604 A048562 A022499 * A134721 A002707 A283247

Adjacent sequences: A129206 A129207 A129208 * A129210 A129211 A129212

Keyword

sign

Author

Michael Somos, Apr 03 2007, Apr 17 2007

Status

approved