A004760 List of numbers whose binary expansion does not begin 10.

0, 1, 3, 6, 7, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• base

Comments

For n >= 2 sequence {a(n+2)} is the minimal recursive such that A007814(a(n+2))=A007814(n). - Vladimir Shevelev, Apr 27 2009

A053645(a(n)) = n-1 for n > 0. - Reinhard Zumkeller, May 20 2009

a(n+1) is also the number of nodes in a complete binary tree with n nodes in the bottommost level. - Jacob Jona Fahlenkamp, Feb 01 2023

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009. [_Vladimir Shevelev_, Apr 15 2009]

Crossrefs

Cf. A004761, A007814, A010060, A159559, A159560, A159615, A159619, A159629, A159698, A004759.

Programs

Maple

0,1,seq(seq(3*2^d+x,x=0..2^d-1),d=0..6); # Robert Israel, Aug 03 2016

Mathematica

Select[Range@ 125, If[Length@ # < 2, #, Take[#, 2]] &@ IntegerDigits[#, 2] != {1, 0} &] (* Michael De Vlieger, Aug 02 2016 *)

PARI

is(n)=n<2 || binary(n)[2] \\ Charles R Greathouse IV, Sep 23 2012

PARI

print1("0, 1");for(i=0,5,for(n=3<Charles R Greathouse IV, Sep 23 2012

PARI

a(n) = if(n<=2,n-1, (n-=2) + 2<Kevin Ryde, Jul 22 2022

Python

def A004760(n): return m+(1<0 else n-1 # Chai Wah Wu, Jul 26 2023

R

maxrow <- 8 # by choice
b01 <- 1
for(m in 0:maxrow){
  b01 <- c(b01,rep(1,2^(m+1))); b01[2^(m+1):(2^(m+1)+2^m-1)] <- 0
}
a <- which(b01 == 1)
# Yosu Yurramendi, Mar 30 2017

Formula

For n > 0, a(n) = 3n - 2 - A006257(n-1). - Ralf Stephan, Sep 16 2003

a(0) = 0, a(1) = 1, for n > 0: a(2n) = 2*a(n) + 1, a(2n+1) = 2*a(n+1). - Philippe Deléham, Feb 29 2004

For n >= 3, A007814(a(n)) = A007814(n-2). - Vladimir Shevelev, Apr 15 2009

a(n+2) = min{m>a(n+1): A007814(m)=A007814(n)}; A010060(a(n+2)) = 1-A010060(n). - Vladimir Shevelev, Apr 27 2009

a(1)=0, a(2)=1, a(2^m+k+2) = 2^(m+1) + 2^m+k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Jul 30 2016

G.f.: x/(1-x)^2 + (x/(1-x))*Sum_{k>=0} 2^k*x^(2^k). - Robert Israel, Aug 03 2016

a(2^m+k) = A004761(2^m+k) + 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

For n > 0, a(n+1) = n + 2^ceiling(log_2(n)) - 1. - Jacob Jona Fahlenkamp, Feb 01 2023

Extensions

Offset changed to 1, b-file corrected. - N. J. A. Sloane, Aug 07 2016