A155589
a(n) = 6^n + 2^n - 1.
Original entry on oeis.org
1, 7, 39, 223, 1311, 7807, 46719, 280063, 1679871, 10078207, 60467199, 362799103, 2176786431, 13060702207, 78364180479, 470185017343, 2821109972991, 16926659575807, 101559956930559, 609359740534783, 3656158441111551, 21936950642475007, 131621703846461439
Offset: 0
Original entry on oeis.org
1, 12, 124, 1338, 14656, 161082, 1771624, 19487298, 214359136, 2357948202, 25937425624, 285311672658, 3138428380816, 34522712152122, 379749833599624, 4177248169448418, 45949729863637696, 505447028499424842
Offset: 0
A143960
a(n) = the n-th positive integer with exactly n zeros and n ones in its binary representation.
Original entry on oeis.org
2, 10, 38, 142, 542, 2110, 8318, 33022, 131582, 525310, 2099198, 8392702, 33562622, 134234110, 536903678, 2147549182, 8590065662, 34360000510, 137439477758, 549756862462, 2199025352702, 8796097216510, 35184380477438, 140737505132542, 562949986975742
Offset: 1
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.
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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 *)
A130567
Expansion of x*(2 - 7*x + 2*x^2)/((1-x)*(1-4*x)*(1-2*x)).
Original entry on oeis.org
2, 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879
Offset: 1
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f[n_Integer?Positive] := f[n] = 2^(2*n - 1) + 2*f[n - 1] + 1; f[0] = 2; Table[f[n], {n, 0, 30}]
CoefficientList[Series[x*(2-7x+2x^2)/((1-x)(1-4x)(1-2x)),{x,0,30}],x] (* Harvey P. Dale, Sep 07 2015 *)