A203625 Indices of octagonal numbers which are also decagonal.

1, 135, 26125, 5068051, 983175705, 190731018655, 37000834443301, 7177971150981675, 1392489402456001585, 270135766105313325751, 52404946135028329194045, 10166289414429390550318915, 1972207741453166738432675401, 382598135552499917865388708815

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*(1-60*x-5*x^2) / ((1-x)*(1-194*x+x^2)).

a(n) = 194*a(n-1)-a(n-2)-64.

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

a(n) = 1/24*((1+2*sqrt(3))*(2+sqrt(3))^(4*n-3)+(1-2*sqrt(3))*(2-sqrt(3))^(4*n-3)+8).

a(n) = ceiling(1/24*(1+2*sqrt(3))*(2+sqrt(3))^(4*n-3)).

Example

The second octagonal number that is also decagonal is A000567(135) = 54405. Hence a(2) = 135.

Mathematica

LinearRecurrence[{195, -195, 1}, {1, 135, 26125}, 14]

Crossrefs

Cf. A203624, A203626, A001107, A000567.

Sequence in context: A132054 A273440 A106175 * A051307 A056740 A065663

Adjacent sequences: A203622 A203623 A203624 * A203626 A203627 A203628

Keyword

nonn,easy

Author

Ant King, Jan 05 2012

Status

approved