1,2

a(n)=(A001835(n))^2. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 21 2006

Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.det), May 12 2009: The equation a(t)*(3*a(t)-2)=m*m is equivalent to the Pell equation (3*a(t)-1)*(3*a(t)-1)-3*m*m=1.

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

Nearest integer to 1/6 * (2+sqrt(3))^(2n-1). - Ralf Stephan, Feb 24 2004

Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.det), May 12 2009: (Start)

With A=(2+sqrt(3))^2=7+4*sqrt(3) the equation x*x-3*m*m=1 has solutions

x(t)+sqrt(3)*m(t)=(2+sqrt(3))*A^t and the recurrences

x(t+2)=14*x(t+1)-x(t) with <x(t)> = 2,26,362,5042

m(t+2)=14*m(t+1)-m(t) with <m(t)> = 1,15,209,2911

a(t+2)=14*a(t+1)-a(t)-4 with <a(t)> = 1,9,121, as above

(End)

Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.det), May 12 2009: (Start)

for n from 1 to 10000 do m=sqrt(3*n*n-2*n): if (trunc(m)=m) then print(n, m):

end if: end do: (End)

Cf. A028230, A036428.

a(n) = A045899(n-1) + 1 = A051047(n+1) + 1 = A003697(2n-2).

Sequence in context: A167722 A103930 A138978 this_sequence A084769 A050353 A112941

Adjacent sequences: A046181 A046182 A046183 this_sequence A046185 A046186 A046187

nonn

Eric Weisstein (eric(AT)weisstein.com)