1,2

the 2 equations are equivalent to the pell-equation x(n)^2-7*y(n)^2=9

with x(n)=7*m(n)+4 and y(n)=a(n)*b(n);

the pell-equation r^2-7*u^2=1 has the smallest integer solutions r=8; u=3;

a(n+4)=16*a(n+2)-a(n); a(1)=1; a(-n)=a(n+1); a(2)=5;

b(n+4)=16*b(n+2)-b(n); b(1)=1; b(-n)=-b(n+1); b(2)=13;

x(n+4)=254*x(n+2)-x(n); x(1)=4; x(-n)=x(n+1); x(2)=172;

y(n+4)=254*y(n+2)-y(n); y(1)=1; y(-n)=-y(n+1); y(2)=65;

m(n+6)=255*(m(n+4)-m(n+2))+m(n); m(1)=0; m(-n)=m(n+1); m(2)=24; m(3)=120;

w:=sqrt(7); 16=2*r; 254=4*r^2-2; 255=4*r^2-1;

a(2*n)=((w-1)*(8+3*w)^n+(w+1)*(8-3*w)^n)/2*w;

a(2*n+1)=((w+1)*(8+3*w)^n+(w-1)*(8-3*w)^n)/2*w;

b(2*n)=((w-1)*(8+3*w)^n-(w+1)*(8-3*w)^n)/2;

b(2*n+1)=((w+1)*(8+3*w)^n-(w-1)*(8-3*w)^n)/2;

x(2*n)=((4-w)*(8+3*w)^(2n)+(4+w)*(8-3*w)^(2n))/2;

x(2*n+1)=((4+w)*(8+3*w)^(2n)+(4-w)+(8-3*w)^(2n))/2;

y(2*n)=((4-w)*(8+3*w)^(2n)-(4+w)*(8-3*w)^(2n))/2*w;

y(2*n+1)=((4+w)*(8+3*w)^(2n)-(4-w)*(8-3*w)^(2n))/2*w;

n=0: for a from 1 to 1000000 do b:=sqrt(7*a^2-6):

if (trunc(b)=b) then n:=n+1: m:=a^2-1: x:=7*m+4: y:=a*b:

print(n, a, b, m, x, y): end if: end do:

Sequence in context: A065555 A067890 A154797 this_sequence A002359 A090518 A057726

Adjacent sequences: A161849 A161850 A161851 this_sequence A161853 A161854 A161855

nonn,uned

Weisenhorn Paul (paulweisenhorn(AT)online.de), Jun 20 2009