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A203624
Numbers which are both decagonal and octagonal.
2
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).
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
KEYWORD
nonn,easy
AUTHOR
Ant King, Jan 05 2012
STATUS
approved