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A203627
Numbers which are both 9-gonal (nonagonal) and 10-gonal (decagonal).
3
1, 1212751, 977965238701, 788633124418157851, 635955328796073362530201, 512835649051022518566661395751, 413551693065406705688396809494274501, 333488912390817262631483541451235285166451, 268926125929366270527488184087670639619302551601
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).
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
LinearRecurrence[{806403, -806403, 1}, {1, 1212751, 977965238701}, 9]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ant King, Jan 06 2012
STATUS
approved