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A203627 Numbers which are both 9-gonal (nonagonal) and 10-gonal (decagonal). 2

%I

%S 1,1212751,977965238701,788633124418157851,635955328796073362530201,

%T 512835649051022518566661395751,413551693065406705688396809494274501,

%U 333488912390817262631483541451235285166451,268926125929366270527488184087670639619302551601

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

%C 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).

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

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

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

%F 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).

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

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

%t LinearRecurrence[{806403, -806403, 1}, {1, 1212751, 977965238701}, 9]

%Y Cf. A203628, A203629, A001107, A001106.

%K nonn,easy

%O 1,2

%A _Ant King_, Jan 06 2012

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Last modified July 31 15:35 EDT 2014. Contains 245085 sequences.