A062880 Zero together with the numbers which can be written as a sum of distinct odd powers of 2.

0, 2, 8, 10, 32, 34, 40, 42, 128, 130, 136, 138, 160, 162, 168, 170, 512, 514, 520, 522, 544, 546, 552, 554, 640, 642, 648, 650, 672, 674, 680, 682, 2048, 2050, 2056, 2058, 2080, 2082, 2088, 2090, 2176, 2178, 2184, 2186, 2208, 2210, 2216, 2218, 2560, 2562

Offset: 0

Author

Antti Karttunen, Jun 26 2001

Keywords

• nonn
• easy

Comments

Binary expansion of n does not contain 1-bits at even positions.

Integers whose base-4 representation consists of only 0's and 2's.

Every nonnegative even number is a unique sum of the form a(k)+2*a(l); moreover, this sequence is unique with such property. - Vladimir Shevelev, Nov 07 2008

Also numbers such that the digital sum base 2 and the digital sum base 4 are in a ratio of 2:4. - Michel Marcus, Sep 23 2013

From Gus Wiseman, Jun 10 2020: (Start)

Numbers k such that the k-th composition in standard order has all even parts. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. For example, the sequence of all compositions into even parts begins:

0: () 520: (6,4) 2080: (6,6)

2: (2) 522: (6,2,2) 2082: (6,4,2)

8: (4) 544: (4,6) 2088: (6,2,4)

10: (2,2) 546: (4,4,2) 2090: (6,2,2,2)

32: (6) 552: (4,2,4) 2176: (4,8)

34: (4,2) 554: (4,2,2,2) 2178: (4,6,2)

40: (2,4) 640: (2,8) 2184: (4,4,4)

42: (2,2,2) 642: (2,6,2) 2186: (4,4,2,2)

128: (8) 648: (2,4,4) 2208: (4,2,6)

130: (6,2) 650: (2,4,2,2) 2210: (4,2,4,2)

136: (4,4) 672: (2,2,6) 2216: (4,2,2,4)

138: (4,2,2) 674: (2,2,4,2) 2218: (4,2,2,2,2)

160: (2,6) 680: (2,2,2,4) 2560: (2,10)

162: (2,4,2) 682: (2,2,2,2,2) 2562: (2,8,2)

168: (2,2,4) 2048: (12) 2568: (2,6,4)

170: (2,2,2,2) 2050: (10,2) 2570: (2,6,2,2)

512: (10) 2056: (8,4) 2592: (2,4,6)

514: (8,2) 2058: (8,2,2) 2594: (2,4,4,2)

(End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance, On the binary expansions of algebraic numbers, J. Théor. Nombres Bordeaux, 16 (2004), 487-518.
• S. Eigen, A. Hajian, and S. Kalikow, Ergodic transformations and sequences of integers, Israel J. Math. 75 (1991), 119-128; Math. Rev. 1147294 (93c:28014).

Crossrefs

Cf. A000695.

Except for first term, n such that A063694(n) = 0. Binary expansion is given in A062033.

Interpreted as Zeckendorf expansion: A062879.

Central diagonal of arrays A163357 and A163359.

Even partitions are counted by A035363.

Numbers with an even number of 1's in binary expansion are A001969.

Numbers whose binary expansion has even length are A053754.

All of the following pertain to compositions in standard order (A066099):

- Length is A000120.

- Compositions without even parts are A060142.

- Sum is A070939.

- Product is A124758.

- Strict compositions are A233564.

- Heinz number is A333219.

- Number of distinct parts is A334028.

Programs

C

uint32_t a_next(uint32_t a_n) { return (a_n + 0x55555556) & 0xaaaaaaaa; } /* Falk Hüffner, Jan 22 2022 */

Haskell

a062880 n = a062880_list !! n
a062880_list = filter f [0..] where
   f 0 = True
   f x = (m == 0 || m == 2) && f x'  where (x', m) = divMod x 4
-- Reinhard Zumkeller, Nov 20 2012

Maple

[seq(a(j),j=0..100)]; a := n -> add((floor(n/(2^i)) mod 2)*(2^((2*i)+1)),i=0..floor_log_2(n+1));

Mathematica

b[n_] := BitAnd[n, Sum[2^k, {k, 0, Log[2, n] // Floor, 2}]]; Select[Range[ 0, 10^4], b[#] == 0&] (* Jean-François Alcover, Feb 28 2016 *)

Python

def A062880(n): return int(bin(n)[2:],4)<<1 # Chai Wah Wu, Aug 21 2023

Formula

a(n) = 2 * A000695(n). - Vladimir Shevelev, Nov 07 2008

From Robert Israel, Apr 10 2018: (Start)

a(2*n) = 4*a(n).

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

G.f. g(x) satisfies: g(x) = 4*(1+x)*g(x^2)+2*x/(1-x^2). (End)