A203627 - OEIS

A203627

Numbers which are both 9-gonal (nonagonal) and 10-gonal (decagonal).

1, 1212751, 977965238701, 788633124418157851, 635955328796073362530201, 512835649051022518566661395751, 413551693065406705688396809494274501, 333488912390817262631483541451235285166451, 268926125929366270527488184087670639619302551601

(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))^8 = 403201+107760*sqrt(14).

LINKS

Table of n, a(n) for n=1..9.

Index entries for linear recurrences with constant coefficients, signature (806403, -806403, 1).

FORMULA

G.f.: x*(1+406348*x+451*x^2) / ((1-x)*(1-806402*x+x^2)).

a(n) = 806402*a(n-1)-a(n-2)+406800.

a(n) = 806403*a(n-1)-806403*a(n-2)+a(n-3).

a(n) = 1/448*((15+2*sqrt(14))*(2*sqrt(2)+sqrt(7))^(8*n-6)+(15-2*sqrt(14))*(2*sqrt(2)-sqrt(7))^(8*n-6)-226).

a(n) = floor(1/448*(15+2*sqrt(14))*(2*sqrt(2)+sqrt(7))^(8*n-6)).

EXAMPLE

The second number that is both nonagonal and decagonal is A001106(589) = A001107(551) = 1212751. Hence a(2) = 1212751.

MATHEMATICA