A129206 - OEIS

%I #18 Nov 14 2019 21:29:52

%S 0,1,-12,-2257,1494696,8914433905,-178761481355556,

%T -62419747600438859233,-5354229862821602092291248,

%U 1001926359199672697329083442936609,50016678000996026579336936742637753055940

%N First 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="/A129206/b129206.txt">Table of n, a(n) for n = 0..49</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/wxyz.gp">Weierstrass Elliptic Function Polynomials (PARI/GP .gp version)</a> or <a href="http://grail.eecs.csuohio.edu~/somos/wxyz.wl">(Mathematica .wl version)</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 W(2*n) = (-1)^n * 2 * W(n) * X(n) * Y(n) * Z(n) for all n in Z.

%F 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.

%o (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) ))};

%Y Cf. A129207, A129208, A129209.

%K sign

%O 0,3

%A _Michael Somos_, Apr 03 2007