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Third sequence in solution to congruent number 5 problem.
4

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

%S 1,2,41,1562,3344161,-7118599318,654686219104361,

%T -128615821825334210638,249850594047271558364480641,

%U -1935878334514951131830244285524398,160443526614433014168714029147613242401001

%N Third 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="/A129208/b129208.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 Y(2*n) = Y(n)^4 + 25 * 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, [2, 41, 1562][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