A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.
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
Offset: 0
Examples
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
References
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 21.
- Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.8 Binary, Octal, Hexadecimal, p. 64.
- 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)
- 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.
- 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.
- G. W. Leibniz, Explication de l'arithmétique binaire, qui se sert des seuls caractères 0 & 1; avec des remarques sur son utilité, et sur ce qu'elle donne le sens des anciennes figures chinoises de Fohy, Mémoires de l'Académie Royale des Sciences, 1703, pp. 85-89; reprinted in Gumin (1979).
- N. J. A. Sloane, Table of a(n) for n = 0..1048576 (A large file).
- Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.
- Index entries for sequences related to Most Wanted Primes video.
- Index entries for 10-automatic sequences.
- Index entries for sequences related to binary expansion of n.
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).
Programs
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Haskell
a007088 0 = 0 a007088 n = 10 * a007088 n' + m where (n',m) = divMod n 2 -- Reinhard Zumkeller, Jan 10 2012
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Maple
A007088 := n-> convert(n, binary): seq(A007088(n), n=0..50); # R. J. Mathar, Aug 11 2009
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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 *) -
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
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Python
def a(n): return int(bin(n)[2:]) print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 10 2021
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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
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(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
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