OFFSET
1,2
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,4468995,0,-4468995,0,1).
FORMULA
G.f.: h(z)=(z*(1+70*z+2000860*z^2+2146726*z^3+148995*z^4+19352060*z^5))/((1-z^2)*(1-4468994*z^2+z^4)).
The subsequences of the even-indexed and odd-indexed terms have the same recurrence relation : a(n+2)=4468994*a(n+1)-a(n)+2149856. The g.f. for the first is f(z)=(z*(1+2000860*z+148995*z^2))/((1-z)*(1-4468994*z+z^2)) and for the second g(z)=(z*(70+2146726*z+19352060*z^2))/((1-z)*(1-4468994*z+z^2)).
Therefore h(z)=(1/z)*f(z^2)+g(z^2).
The problem of pentagonal 13-gonal numbers is connected with the Diophantine equation 3*X^2=11*Y^2+232 where X=22*p-9 and Y=6*q-1 ; the parametrization of the conic gives the other equation 33*X^2=Z^2+2552 and the fact that the solutions fall into two sets.
The first uses the values 1, 1085 and the recurrence a(n+2)=2114*a(n+1)-a(n)-864 for p; the second uses 4,7568 and the recurrence a(n+2)=2114*a(n+1)-a(n)-352 for q.
X and Y satisfy the same recurrence a(n+2)=2114*a(n+1)-a(n).
a(n) = 4468995*a(n-2)-4468995*a(n-4)+a(n-6) for n>6. - Colin Barker, Oct 20 2014
PROG
(PARI) Vec((x*(1+70*x+2000860*x^2+2146726*x^3+148995*x^4+19352060*x^5))/((1-x^2)*(1-4468994*x^2+x^4)) + O(x^20)) \\ Colin Barker, Oct 20 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard Choulet, Jul 18 2007
EXTENSIONS
More terms from Colin Barker, Oct 20 2014
STATUS
approved