A203628 - OEIS

A203628

Indices of 9-gonal (nonagonal) numbers which are also 10-gonal (decagonal).

1, 589, 528601, 474682789, 426264615601, 382785150126589, 343740638549061001, 308678710631906651989, 277193138406813624424801, 248919129610608002826818989, 223529101197187579724859027001, 200728883955944835984920579427589

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (2*sqrt(2)+sqrt(7))^4 = 449+120*sqrt(14).

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (899, -899, 1).

FORMULA

G.f.: x*(1-310*x-11*x^2) / ((1-x)*(1-898*x+x^2)).

a(n) = 898*a(n-1)-a(n-2)-320.

a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).

a(n) = 1/56*((sqrt(2)+2*sqrt(7))*(2*sqrt(2)+sqrt(7))^(4*n-3)+(sqrt(2)-2*sqrt(7))*(2*sqrt(2)-sqrt(7))^(4*n-3)+20).

a(n) = ceiling(1/56*(sqrt(2)+2*sqrt(7))*(2*sqrt(2)+sqrt(7))^(4*n-3)).

EXAMPLE

The second number that is both 9-gonal (nonagonal) and 10-gonal (decagonal) is A001106(589) = 1212751. Hence a(2) = 589.

MATHEMATICA