A203626 Indices of decagonal numbers which are also octagonal.

1, 117, 22625, 4389061, 851455137, 165177907445, 32043662589121, 6216305364381957, 1205931197027510465, 233944435917972648181, 45384014636889666236577, 8804264895120677277247685, 1707982005638774502119814241, 331339704829027132733966714997

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(3))^4 = 97+56*sqrt(3).

Links

Table of n, a(n) for n=1..14.

Index entries for linear recurrences with constant coefficients, signature (195, -195, 1).

Formula

G.f.: x*(1-78*x+5*x^2) / ((1-x)*(1-194*x+x^2)).

a(n) = 194*a(n-1)-a(n-2)-72.

a(n) = 195*a(n-1)-195*a(n-2)+a(n-3).

a(n) = 1/48*((sqrt(3)+6)*(2+sqrt(3))^(4*n-3)-(sqrt(3)-6)*(2-sqrt(3))^(4*n-3)+18).

a(n) = ceiling(1/48*(sqrt(3)+6)*(2+sqrt(3))^(4*n-3)).

Example

The second decagonal number that is also octagonal is A001107(117) = 54405. Hence a(2) = 117.

Mathematica

LinearRecurrence[{195, -195, 1}, {1, 117, 22625}, 14]

Crossrefs

Cf. A203624, A203625, A001107, A000567.

Sequence in context: A223749 A218416 A027511 * A220600 A194613 A079196

Adjacent sequences: A203623 A203624 A203625 * A203627 A203628 A203629

Keyword

nonn,easy

Author

Ant King, Jan 05 2012

Status

approved