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A038573 2^A000120(n)-1. 18
0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Essentially the same sequence as A001316, which has much more information. - N. J. A. Sloane, Jun 05 2009

Smallest number with same number of 1's in its binary expansion as n.

Fixed point of the morphism 0 -> 01, 1 -> 13, 3 -> 37, ... = k -> k, 2k+1, ... starting from a(0) = 0; 1 -> 01 -> 0113 -> 01131337 -> 011313371337377(15) -> ..., . - Robert G. Wilson v Jan 24 2006. ...........

Contribution from Gary W. Adamson, Jun 04 2009: (Start)

As an infinite string, 2^n terms per row starting with "1":

(1; 1,3; 1,3,3,7; 1,3,3,7,3,7,7,15; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,3l;...)

Row sums of that triangle = A027649: (1, 4, 14, 46, 454,...); where the

next row sum = current term of A027649 + next term in finite difference

row of A027649, i.e. (1, 3, 10, 32, 100, 308,...) = A053581. (End)

LINKS

T. D. Noe, Table of n, a(n) for n=0..1023

Michael Gilleland, Some Self-Similar Integer Sequences

FORMULA

a(2n) = a(n), a(2n+1) = 2*a(n)+1, a(0) = 0. a(n) = A001316(n)-1 = 2^A000120(n)-1 (comment from Daniele Parisse (daniele.parisse(AT)m.dasa.de)).

a(n) = number of positive integers k < n such that n XOR k = n-k (cf. A115378). - Paul D. Hanna, Jan 21 2006

a(n) = f(n, 1) with f(x, y) = if x = 0 then y - 1 else f(floor(x/2), y*(1 + x mod 2)). [From Reinhard Zumkeller, Nov 21 2009]

a(n) = (n mod 2 + 1) * a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Oct 10 2012

EXAMPLE

9 = 1001 -> 0011 -> 3, so a(9)=3.

Contribution from Gary W. Adamson, Jun 04 2009: (Start)

Triangle read by rows:

. 1;

. 1, 3;

. 1, 3, 3, 7;

. 1, 3, 3, 7, 3, 7, 7, 15;

. 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31;

. ...

Row sums: (1, 4, 14, 46,...) = A026749 = last row terms + new set of terms

such that row 3 = (1, 3, 3, 7,) + (3, 7, 7, 15) = 14 + 32 = A027649(3) + A053581(3). (End)

The rows of this triangle converge to A159913. - N. J. A. Sloane, Jun 05 2009

MATHEMATICA

Array[ 2^Count[ IntegerDigits[ #, 2 ], 1 ]-1&, 100 ]

Nest[ Flatten[ # /. a_Integer -> {a, 2a + 1}] &, {0}, 7] (from Robert G. Wilson v, Jan 24 2006)

PROG

(PARI) a(n)=2^subst(Pol(binary(n)), x, 1)-1

(Haskell)

a038573 0 = 0

a038573 n = (m + 1) * (a038573 n') + m where (n', m) = divMod n 2

-- Reinhard Zumkeller, Oct 10 2012, Feb 07 2011

CROSSREFS

Cf. A007313, A115378.

This is Guy Steele's sequence GS(3, 6) (see A135416).

A027649, A053581 [From Gary W. Adamson, Jun 04 2009]

Cf. A000079. [From Omar E. Pol, Jun 07 2009]

Sequence in context: A005885 A205145 A061892 * A173465 A151837 A163381

Adjacent sequences:  A038570 A038571 A038572 * A038574 A038575 A038576

KEYWORD

nonn,easy,nice

AUTHOR

Marc LeBrun

EXTENSIONS

More terms from Erich Friedman.

New definition from N. J. A. Sloane, Mar 01 2008

STATUS

approved

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Last modified August 29 04:05 EDT 2014. Contains 246185 sequences.