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%I #13 Mar 28 2024 09:01:50
%S 1,12,850,16761,1225159,24168810,1766677888,34851406719,2547548288797,
%T 50255704319448,3673562865766846,72468690777236757,
%U 5297275104887502595,104499801845071083606,7638667027684912974604,150688641791901725322555,11014952556646539621875833
%N Indices of decagonal numbers that are also heptagonal.
%C As n increases, the ratios of consecutive terms settle into an approximate 2-cycle with a(n)/a(n-1) bounded above and below by 1/9*(329+104*sqrt(10)) and 1/9*(89+28*sqrt(10)) respectively.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1442,-1442,-1,1).
%F G.f.: x(1+11*x-604*x^2+49*x^3+3*x^4) / ((1-x)*(1-38*x+x^2)*(1+38*x+x^2)).
%F a(n) = 1442*a(n-2)-a(n-4)-540.
%F a(n) = a(n-1)+1442*a(n-2)-1442*a(n-3)-a(n-4)+a(n-5).
%F a(n) = 1/80*((sqrt(10)-(-1)^n)*(5-sqrt(10))* (3+sqrt(10))^(2*n-1)-(sqrt(10)+(-1)^n)*(5+sqrt(10))*(3-sqrt(10))^(2*n-1)+30).
%F a(n) = ceiling(1/80*(sqrt(10)-(-1)^n)*(5-sqrt(10))*(3+sqrt(10))^(2*n-1))
%e The second decagonal number that is also heptagonal is A001107(12)=540. Hence a(2)=12.
%t LinearRecurrence[{1, 1442, -1442, -1, 1}, {1, 12, 850, 16761, 1225159}, 17]
%Y Cf. A203408, A203409, A001107, A000566.
%K nonn,easy
%O 1,2
%A _Ant King_, Jan 03 2012
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