A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

2, 9, 35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135, 140737496743935, 562949970198527

Offset

1,1

Comments

Smallest numbers having in binary representation n 0's and n 1's: a(n) = Min{m: A023416(m)=A000120(m)=n}.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Eric Weisstein's World of Mathematics, Binary

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Formula

a(n+1) = 2*a(n) + 4^n + 1.

a(n) = 2^(2*n-1) + 2^(n-1) - 1.

a(n) = A007582(n)-1 = A056326(2n+1) = A005367(n-1)/2 = A063376(n)/2-1 = A032125(n+1)/3-1 = A056309(2n+1)/2 = A028403(n+1)/4-1 = (A001576(n)-3)/2 = (A028400(n+1)-9)/8 = Sum_{k=2..n+1} A049775(k). - Ralf Stephan, Mar 24 2004

G.f.: x*(-2+5*x) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011

E.g.f.: exp(x)*(exp(3*x) + exp(x) - 2)/2. - Stefano Spezia, Sep 27 2023

Mathematica

LinearRecurrence[{7, -14, 8}, {2, 9, 35}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
Table[FromDigits[Join[PadRight[{1}, n, 0], PadRight[{}, n-2, 1]], 2], {n, 2, 30}]//Sort (* or *) Rest[CoefficientList[Series[x (-2+5x)/((x-1)(2x-1)(4x-1)), {x, 0, 30}], x]] (* Harvey P. Dale, Jul 30 2021 *)

Crossrefs

Cf. A007088, A001700.

Subsequence of A031443.

Cf. A000120, A023416.

Cf. A007582, A056326, A005367, A063376, A032125, A056309, A028403, A001576, A028400, A049775.

Sequence in context: A255410 A083141 A001571 * A147762 A077837 A289974

Adjacent sequences: A092428 A092429 A092430 * A092432 A092433 A092434

Keyword

nonn,easy

Author

Reinhard Zumkeller, Mar 23 2004

Status

approved