A129209 - OEIS

%I #10 May 28 2018 11:37:22

%S 1,3,31,5283,-113279,21166249443,-518493692732129,

%T 189797666150873887683,-249563579992463717493803519,

%U 2960896329211804542556804051252803,-115038188620995226180802686473825513089249

%N Fourth sequence in solution to congruent number 5 problem.

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

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

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

%H Seiichi Manyama, <a href="/A129209/b129209.txt">Table of n, a(n) for n = 0..49</a>

%F 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.

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

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

%o (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) ))};

%K sign

%O 0,2

%A _Michael Somos_, Apr 03 2007, Apr 17 2007