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A203628
Indices of 9-gonal (nonagonal) numbers which are also 10-gonal (decagonal).
2
1, 589, 528601, 474682789, 426264615601, 382785150126589, 343740638549061001, 308678710631906651989, 277193138406813624424801, 248919129610608002826818989, 223529101197187579724859027001, 200728883955944835984920579427589
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).
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
LinearRecurrence[{899, -899, 1}, {1, 589, 528601}, 12]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ant King, Jan 06 2012
STATUS
approved