login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007088 The binary numbers: numbers written in base 2.
(Formerly M4679)
551
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

a(A000290(n)) = A001737(n). - Reinhard Zumkeller, Apr 25 2009

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 > 0: A054055(a(n)) = 1. - Reinhard Zumkeller, Apr 25 2012

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

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.

G. W. Leibniz, Explication de l'arithmétic binaire, qui se sert des seules caracteres 0 & 1; avec des remarques sur son utilité, & sur ce qu'elle donne le sens des anciennes Chinoises de Fohy, Memoires de l'Academie Royale des Sciences, 1703, pp. 85-89; reprinted in Gumin (1979).

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

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..8192

N. J. A. Sloane, Table of a(n) for n = 0..1048576

Index entries for 10-automatic sequences.

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = Sum{d(i)*10^i: i=0, 1, ..., m}, where Sum{d(i)*2^i: i=0, 1, ..., m} 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

a(n) = Sum_{k=0..floor(log2(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 + ...

MAPLE

A007088 := proc(n) local dgs ; dgs := convert(n, base, 2) ; add( op(i, dgs)*10^(i-1), i=1..nops(dgs)) ; end: # 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 *)

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

CROSSREFS

The basic sequences concerning the binary expansion of n are this one, A000788, A000069, A001969, A023416, A059015, A000120. Bisections A099820 and A099821.

Cf. A000042, A007089-A007095, A000695, A005836, A033042-A033052, A159918, A004290, A169965, A169966, A169967, A169964, A204093, A204094, A204095, A097256, A257773, A257770.

Sequence in context: A266946 A081551 A257831 * A115848 A136814 A136809

Adjacent sequences:  A007085 A007086 A007087 * A007089 A007090 A007091

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 22 15:30 EDT 2017. Contains 286876 sequences.