| 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 recurrences 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)
a(n) = +a(n-1) +1442*a(n-3) -1442*a(n-4) -a(n-6) +a(n-7). G.f.: -x*(1+5*x+43*x^2-530*x^3+43*x^4+5*x^5+x^6) / ( (x-1)*(x^6-1442*x^3+1) ). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 01 2010]
a(n) = 1442*a(n-3) - a(n-6) - 432. - Ant King, Nov 12 2011
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