A006364 Numbers k with an even number of 1's in binary, ignoring last bit. (Formerly M4060)

0, 1, 6, 7, 10, 11, 12, 13, 18, 19, 20, 21, 24, 25, 30, 31, 34, 35, 36, 37, 40, 41, 46, 47, 48, 49, 54, 55, 58, 59, 60, 61, 66, 67, 68, 69, 72, 73, 78, 79, 80, 81, 86, 87, 90, 91, 92, 93, 96, 97, 102, 103, 106, 107, 108, 109, 114, 115, 116, 117, 120, 121, 126, 127, 130, 131, 132

Offset

1,3

Comments

Equivalently, numbers k such that k has an odd number of 1's in binary if and only if k is odd. - Aaron Weiner, Jun 19 2013

Numbers k with an even number of trailing zeros in the binary representation of k!, A011371(k). - Amiram Eldar, Sep 05 2024

References

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 111.

R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

Index entries for sequences related to binary expansion of n.

Formula

Union of 2*A001969 and 2*A001969+1. With initial index 0: a(2n+1) = a(2n)+1, a(4n) = a(2n)+4n, a(4n+2) = -a(2n)+12n+6. - Ralf Stephan, Oct 17 2003

Conjecture: a(n) = 2*n + (-1)^(n-A000120(n-1)) - (3+(-1)^n)/2. - Velin Yanev, Dec 21 2016

Example

G.f. = x + 6*x^2 + 7*x^3 + 10*x^4 + 11*x^5 + 12*x^6 + 13*x^7 + 18*x^8 + ...

Mathematica

Select[Range[0, 150], EvenQ[Count[Most[IntegerDigits[#, 2]], 1]]&] (* Harvey P. Dale, Nov 03 2011 *)
a[ n_] := Which[ n < 1, 0, Mod[n, 2] > 0, a[n - 1] + 1, Mod[n, 4] > 0, 3 n - a[n/2 - 1], True, n + a[n/2]]; (* Michael Somos, Dec 21 2016 *)

Prog

(PARI) a(n)=if(n<1, 0, if(n%2==0, if(n%4==0, a(n/2)+n, -a((n-2)/2)+3*n), a(n-1)+1)) \\ Ralf Stephan
(PARI) is(n)=hammingweight(n>>1)%2==0 \\ Charles R Greathouse IV, Jun 19 2013
(Haskell)
a006364 n = a006364_list
a006364_list = filter (even . a000120. (`div` 2)) [0..]
-- Reinhard Zumkeller, Oct 03 2011

Crossrefs

Cf. A000120, A001969, A011371.

Sequence in context: A286473 A358432 A165363 * A037301 A163247 A085267

Adjacent sequences: A006361 A006362 A006363 * A006365 A006366 A006367

Keyword

base,nonn,nice,easy

Author

N. J. A. Sloane

Status

approved