OFFSET
0,2
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).
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)
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 *)
PROG
(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
(C) uint32_t a_next(uint32_t a_n) { return (a_n + 0x55555556) & 0xaaaaaaaa; } /* Falk Hüffner, Jan 22 2022 */
(Python)
def A062880(n): return int(bin(n)[2:], 4)<<1 # Chai Wah Wu, Aug 21 2023
CROSSREFS
Cf. A000695.
Interpreted as Zeckendorf expansion: A062879.
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.
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, Jun 26 2001
STATUS
approved