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 A203625 Indices of octagonal numbers which are also decagonal. 2

%I

%S 1,135,26125,5068051,983175705,190731018655,37000834443301,

%T 7177971150981675,1392489402456001585,270135766105313325751,

%U 52404946135028329194045,10166289414429390550318915,1972207741453166738432675401,382598135552499917865388708815

%N Indices of octagonal numbers which are also decagonal.

%C 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).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195, -195, 1).

%F G.f.: x*(1-60*x-5*x^2) / ((1-x)*(1-194*x+x^2)).

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

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

%F 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).

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

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

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

%Y Cf. A203624, A203626, A001107, A000567.

%K nonn,easy

%O 1,2

%A _Ant King_, Jan 05 2012

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Last modified January 29 11:06 EST 2023. Contains 359922 sequences. (Running on oeis4.)