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

1,3

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]

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

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 *)

Prog

(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<<i, (4<<i)-1, print1(", "n))) \\ Charles R Greathouse IV, Sep 23 2012
(PARI) a(n) = if(n<=2, n-1, (n-=2) + 2<<logint(n, 2)); \\ Kevin Ryde, Jul 22 2022
(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
(Python)
def A004760(n): return m+(1<<m.bit_length()) if (m:=n-2)>0 else n-1 # Chai Wah Wu, Jul 26 2023

Crossrefs

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

Sequence in context: A028802 A141742 A004755 * A186083 A093906 A332812

Adjacent sequences: A004757 A004758 A004759 * A004761 A004762 A004763

Keyword

nonn,easy,base

Author

N. J. A. Sloane

Extensions

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

Status

approved