1,2

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), May 01 2009: (Start)

Pell equations: r^2-10*s^2=1 with solution (19,6)

(10*n-3)^2-10*(2*m)^2=9; basic solutions: (7,-2); (7,+2)((57,18);

with x=10*n-3; y=2*m; A=(19+6*sqrt(10))^2; B=(19-6*sqrt(10))^2 one get

x(3*k)+sqrt(10)*y(3*k)=(7-2*sqrt(10))*A^k;

x(3*k+1)+sqrt(10)*y(3*k+1)=(7+2*sqrt(10))*A^k;

x(3*k+2)+sqrt(10)*y(3*k+2)=(57+18*sqrt(10))*A^k;

with the eigenwerten A=721+228*sqrt(10); B=721-228*sqrt(10)

one get the recurences with 1442=4*19*19-2

x(k+6)=1442*x(k+3)-x(k); y(k+6)=1442*y(k+3)-y(k);

m(k+6)=1442*m(k+3)-m(k); n(k+6)=1442*n(k+3)-n(k)-432;

and the explicit formulas

x(3*k+1)=(7*(A^k+B^k)+2*sqrt(10)*(A^k-B^k))/2;

x(3*k+2)=(57*(A^k+B^k)+18*sqrt(10)*(A^k-B^k))/2;

x(3*k)=(7*(A^k+B^k)-2*sqrt(10)*(A^k-B^k))/2;

y(3*k+1)=(7*(A^k-B^k)/sqrt(10)+2*(A^k+B^k)/2;

y(3*k+2)=(57*(A^k-B^k)/sqrt(10)+18*(A^k+B^k))/2;

y(3*k)=(7*(A^k-B^k)/sqrt(10)-2*(A^k+B^k))/2;

n(k)=(x(k)+3)/10; m(k)=y(k)/2;

(End)

Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), May 01 2009: (Start)

for n from 1 to 10000 do m:=sqrt((5*n^2-3*n)/2):

if (trunc(m)=m) then print(n, m): end if: end do:

(End)

Cf. A000566, A036254, A046196.

Sequence in context: A055847 A143165 A008786 this_sequence A024268 A062801 A035290

Adjacent sequences: A046192 A046193 A046194 this_sequence A046196 A046197 A046198

nonn

Eric Weisstein (eric(AT)weisstein.com)