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A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.
(Formerly M4679)
748
0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
List of binary numbers. (This comment is to assist people searching for that particular phrase. - N. J. A. Sloane, Apr 08 2016)
Or, numbers that are sums of distinct powers of 10.
Or, numbers having only digits 0 and 1 in their decimal representation.
Complement of A136399; A064770(a(n)) = a(n). - Reinhard Zumkeller, Dec 30 2007
From Rick L. Shepherd, Jun 25 2009: (Start)
Nonnegative integers with no decimal digit > 1.
Thus nonnegative integers n in base 10 such that kn can be calculated by normal addition (i.e., n + n + ... + n, with k n's (but not necessarily k + k + ... + k, with n k's)) or multiplication without requiring any carry operations for 0 <= k <= 9. (End)
For n > 1: A257773(a(n)) = 10, numbers that are Belgian-k for k=0..9. - Reinhard Zumkeller, May 08 2015
For any integer n>=0, find the binary representation and then interpret as decimal representation giving a(n). - Michael Somos, Nov 15 2015
N is in this sequence iff A007953(N) = A101337(N). A028897 is a left inverse. - M. F. Hasler, Nov 18 2019
For n > 0, numbers whose largest decimal digit is 1. - Stefano Spezia, Nov 15 2023
REFERENCES
Heinz Gumin, "Herrn von Leibniz' 'Rechnung mit Null und Eins'", Siemens AG, 3. Auflage 1979 -- contains facsimiles of Leibniz's papers from 1679 and 1703.
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991, p. 383.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..32768 (first 8192 terms from Franklin T. Adams-Watters)
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
N. J. A. Sloane, Table of a(n) for n = 0..1048576 (A large file)
FORMULA
a(n) = Sum_{i=0..m} d(i)*10^i, where Sum_{i=0..m} d(i)*2^i is the base 2 representation of n.
a(n) = (1/2)*Sum_{i>=0} (1-(-1)^floor(n/2^i))*10^i. - Benoit Cloitre, Nov 20 2001
a(n) = A097256(n)/9.
a(2n) = 10*a(n), a(2n+1) = a(2n)+1.
G.f.: 1/(1-x) * Sum_{k>=0} 10^k * x^(2^k)/(1+x^(2^k)) - for sequence as decimal integers. - Franklin T. Adams-Watters, Jun 16 2006
a(A000290(n)) = A001737(n). - Reinhard Zumkeller, Apr 25 2009
a(n) = Sum_{k>=0} A030308(n,k)*10^k. - Philippe Deléham, Oct 19 2011
For n > 0: A054055(a(n)) = 1. - Reinhard Zumkeller, Apr 25 2012
a(n) = Sum_{k=0..floor(log_2(n))} floor((Mod(n/2^k, 2)))*(10^k). - José de Jesús Camacho Medina, Jul 24 2014
EXAMPLE
a(6)=110 because (1/2)*((1-(-1)^6)*10^0 + (1-(-1)^3)*10^1 + (1-(-1)^1)*10^2) = 10 + 100.
G.f. = x + 10*x^2 + 11*x^3 + 100*x^4 + 101*x^5 + 110*x^6 + 111*x^7 + 1000*x^8 + ...
.
000 The numbers < 2^n can be regarded as vectors with
001 a fixed length n if padded with zeros on the left
010 side. This represents the n-fold Cartesian product
011 over the set {0, 1}. In the example on the left,
100 n = 3. (See also the second Python program.)
101 Binary vectors in this format can also be seen as a
110 representation of the subsets of a set with n elements.
111 - Peter Luschny, Jan 22 2024
MAPLE
A007088 := n-> convert(n, binary): seq(A007088(n), n=0..50); # R. J. Mathar, Aug 11 2009
MATHEMATICA
Table[ FromDigits[ IntegerDigits[n, 2]], {n, 0, 39}]
Table[Sum[ (Floor[( Mod[f/2 ^n, 2])])*(10^n) , {n, 0, Floor[Log[2, f]]}], {f, 1, 100}] (* José de Jesús Camacho Medina, Jul 24 2014 *)
FromDigits/@Tuples[{1, 0}, 6]//Sort (* Harvey P. Dale, Aug 10 2017 *)
PROG
(PARI) {a(n) = subst( Pol( binary(n)), x, 10)}; /* Michael Somos, Jun 07 2002 */
(PARI) {a(n) = if( n<=0, 0, n%2 + 10*a(n\2))}; /* Michael Somos, Jun 07 2002 */
(PARI) a(n)=fromdigits(binary(n), 10) \\ Charles R Greathouse IV, Apr 08 2015
(Haskell)
a007088 0 = 0
a007088 n = 10 * a007088 n' + m where (n', m) = divMod n 2
-- Reinhard Zumkeller, Jan 10 2012
(Python)
def a(n): return int(bin(n)[2:])
print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 10 2021
(Python)
from itertools import product
n = 4
for p in product([0, 1], repeat=n): print(''.join(str(x) for x in p))
# Peter Luschny, Jan 22 2024
CROSSREFS
The basic sequences concerning the binary expansion of n are this one, A000120 (Hammingweight: sum of bits), A000788 (partial sums of A000120), A000069 (A000120 is odd), A001969 (A000120 is even), A023416 (number of bits 0), A059015 (partial sums). Bisections A099820 and A099821.
Cf. A028897 (convert binary to decimal).
Sequence in context: A266946 A081551 A257831 * A115848 A136814 A136809
KEYWORD
nonn,base,nice,easy
AUTHOR
STATUS
approved

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Last modified May 25 07:15 EDT 2024. Contains 372782 sequences. (Running on oeis4.)