A103165 Pentagonal 13-gonal numbers.

1, 70, 6469855, 314976376, 28913745325725, 1407627555984530, 129215354378188750651, 6290679101929235185300, 577462643423970343783639225, 28113007162445732840160102526, 2580677088685733707192642808088855, 125636860330920714709176528014392400

Offset

1,2

Links

Table of n, a(n) for n=1..12.

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

Sequence in context: A228521 A222746 A066789 * A116276 A056666 A172680

Adjacent sequences: A103162 A103163 A103164 * A103166 A103167 A103168

Keyword

nonn,easy

Author

Richard Choulet, Jul 18 2007

Extensions

More terms from Colin Barker, Oct 20 2014

Status

approved