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%I #15 Mar 11 2022 18:27:43
%S 2,10,38,142,542,2110,8318,33022,131582,525310,2099198,8392702,
%T 33562622,134234110,536903678,2147549182,8590065662,34360000510,
%U 137439477758,549756862462,2199025352702,8796097216510,35184380477438,140737505132542,562949986975742
%N a(n) = the n-th positive integer with exactly n zeros and n ones in its binary representation.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F a(n) = 2^(2n-1) + 2^n - 2.
%F G.f.: 2*x*(1-2*x-2*x^2)/((1-x)*(1-4*x)*(1-2*x)). a(n) = 2*A099393(n-1). [_R. J. Mathar_, Nov 03 2008; G.f. corrected by _Georg Fischer_, May 12 2019]
%e The first of the (10) positive integers with exactly three 0's and three 1's in their binary representation are 35 (100011 in binary), 37 (100101 in binary), 38 (100110 in binary), etc. a(3) is the third of these, which is 38.
%t Table[FromDigits[Select[Sort[Permutations[Flatten[Table[{1,0},n]]]],#[[1]] == 1&][[n]],2],{n,25}] (* or *) Table[2^(2n-1)+2^n-2,{n,25}] (* or *) LinearRecurrence[{7,-14,8},{2,10,38},25] (* The second and third programs are much faster than the first. *) (* _Harvey P. Dale_, Mar 11 2022 *)
%Y Cf. A099393.
%K base,nonn
%O 1,1
%A _Leroy Quet_, Sep 05 2008
%E More terms from _R. J. Mathar_, Nov 03 2008