A203629 Indices of 10-gonal (decagonal) numbers which are also 9-gonal (nonagonal).

1, 551, 494461, 444025091, 398734036921, 358062721129631, 321539924840371381, 288742494443932370171, 259290438470726428041841, 232842525004217888449202711, 209092328163349193100955992301, 187764677848162571186770031883251

Offset: 1

Author

Ant King, Jan 06 2012

Keywords

• nonn
• easy

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).

Examples

			The second number that is both 9-gonal (nonagonal) and 10-gonal (decagonal) is A001107(551) = 1212751. Hence a(2) = 551.
		

Links

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

Crossrefs

Cf. A203627, A203628, A001107, A001106.

Programs

Mathematica

LinearRecurrence[{899, -899, 1}, {1, 551, 494461}, 12]

Formula

G.f.: x*(1-348*x+11*x^2) / ((1-x)*(1-898*x+x^2)).

a(n) = 898*a(n-1)-a(n-2)-336.

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

a(n) = 1/112*((sqrt(7)+7*sqrt(2))*(2*sqrt(2)+sqrt(7))^(4*n-3)-(sqrt(7)-7*sqrt(2))*(2*sqrt(2)-sqrt(7))^(4*n-3)+42).

a(n) = ceiling(1/112*(sqrt(7)+7*sqrt(2))*(2*sqrt(2)+sqrt(7))^(4*n-3)).