A203624 Numbers which are both decagonal and octagonal.

1, 54405, 2047494625, 77055412679701, 2899903398740389665, 109134964431140411989765, 4107185248501634866082443201, 154569809532975562119006255453525, 5817080207856817056285046551655533505, 218919996387913643563255879805998092490501

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))^8 = 18817+10864*sqrt(3).

Links

Table of n, a(n) for n=1..10.

Index entries for linear recurrences with constant coefficients, signature (37635, -37635, 1).

Formula

G.f.: x*(1+16770*x+85*x^2) / ((1-x)*(1-37634*x+x^2)).

a(n) = 37634*a(n-1)-a(n-2)+16856.

a(n) = 37635*a(n-1)-37635*a(n-2)+a(n-3).

a(n) = 1/192*((13+4*sqrt(3))*(2+sqrt(3))^(8*n-6)+(13-4*sqrt(3))*(2-sqrt(3))^(8*n-6)-86).

a(n) = floor(1/192*(13+4*sqrt(3))*(2+sqrt(3))^(8*n-6)).

Example

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

Mathematica

LinearRecurrence[{37635, -37635, 1}, {1, 54405, 2047494625}, 10]

Crossrefs

Cf. A203625, A203626, A001107, A000567.

Sequence in context: A202315 A270763 A251479 * A250857 A083616 A235104

Adjacent sequences: A203621 A203622 A203623 * A203625 A203626 A203627

Keyword

nonn,easy

Author

Ant King, Jan 05 2012

Status

approved