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A007088
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The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.
(Formerly M4679)
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739
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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
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OFFSET
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0,3
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COMMENTS
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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.
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 any integer n>=0, find the binary representation and then interpret as decimal representation giving a(n). - Michael Somos, Nov 15 2015
For n > 0, numbers whose largest decimal digit is 1. - Stefano Spezia, Nov 15 2023
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REFERENCES
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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).
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LINKS
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FORMULA
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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(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
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EXAMPLE
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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 + ...
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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.
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MAPLE
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MATHEMATICA
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Table[ FromDigits[ IntegerDigits[n, 2]], {n, 0, 39}]
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PROG
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(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 */
(Haskell)
a007088 0 = 0
a007088 n = 10 * a007088 n' + m where (n', m) = divMod n 2
(Python)
def a(n): return int(bin(n)[2:])
(Python)
from itertools import product
n = 4
for p in product([0, 1], repeat=n): print(''.join(str(x) for x in p))
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CROSSREFS
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Cf. A028897 (convert binary to decimal).
Cf. A000042, A007089-A007095, A000695, A005836, A033042-A033052, A159918, A004290, A169965, A169966, A169967, A169964, A204093, A204094, A204095, A097256, A257773, A257770.
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KEYWORD
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nonn,base,nice,easy
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AUTHOR
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STATUS
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approved
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