cp's OEIS Frontend

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

%I A004760 #64 Jul 27 2023 08:19:18
%S A004760 0,1,3,6,7,12,13,14,15,24,25,26,27,28,29,30,31,48,49,50,51,52,53,54,
%T A004760 55,56,57,58,59,60,61,62,63,96,97,98,99,100,101,102,103,104,105,106,
%U A004760 107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122
%N A004760 List of numbers whose binary expansion does not begin 10.
%C A004760 For n >= 2 sequence {a(n+2)} is the minimal recursive such that A007814(a(n+2))=A007814(n). - _Vladimir Shevelev_, Apr 27 2009
%C A004760 A053645(a(n)) = n-1 for n > 0. - _Reinhard Zumkeller_, May 20 2009
%C A004760 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
%H A004760 Michael De Vlieger, <a href="/A004760/b004760.txt">Table of n, a(n) for n = 1..10000</a>
%H A004760 Vladimir Shevelev, <a href="http://arXiv.org/abs/0904.2101">Several results on sequences which are similar to the positive integers</a>, arXiv:0904.2101 [math.NT], 2009. [_Vladimir Shevelev_, Apr 15 2009]
%F A004760 For n > 0, a(n) = 3n - 2 - A006257(n-1). - _Ralf Stephan_, Sep 16 2003
%F A004760 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
%F A004760 For n >= 3, A007814(a(n)) = A007814(n-2). - _Vladimir Shevelev_, Apr 15 2009
%F A004760 a(n+2) = min{m>a(n+1): A007814(m)=A007814(n)}; A010060(a(n+2)) = 1-A010060(n). - _Vladimir Shevelev_, Apr 27 2009
%F A004760 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
%F A004760 G.f.: x/(1-x)^2 + (x/(1-x))*Sum_{k>=0} 2^k*x^(2^k). - _Robert Israel_, Aug 03 2016
%F A004760 a(2^m+k) = A004761(2^m+k) + 2^m, m >= 0, 0 <= k < 2^m. - _Yosu Yurramendi_, Aug 08 2016
%F A004760 For n > 0, a(n+1) = n + 2^ceiling(log_2(n)) - 1. - _Jacob Jona Fahlenkamp_, Feb 01 2023
%p A004760 0,1,seq(seq(3*2^d+x,x=0..2^d-1),d=0..6); # _Robert Israel_, Aug 03 2016
%t A004760 Select[Range@ 125, If[Length@ # < 2, #, Take[#, 2]] &@ IntegerDigits[#, 2] != {1, 0} &] (* _Michael De Vlieger_, Aug 02 2016 *)
%o A004760 (PARI) is(n)=n<2 || binary(n)[2] \\ _Charles R Greathouse IV_, Sep 23 2012
%o A004760 (PARI) print1("0, 1");for(i=0,5,for(n=3<<i,(4<<i)-1,print1(", "n))) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A004760 (PARI) a(n) = if(n<=2,n-1, (n-=2) + 2<<logint(n,2)); \\ _Kevin Ryde_, Jul 22 2022
%o A004760 (R)
%o A004760 maxrow <- 8 # by choice
%o A004760 b01 <- 1
%o A004760 for(m in 0:maxrow){
%o A004760   b01 <- c(b01,rep(1,2^(m+1))); b01[2^(m+1):(2^(m+1)+2^m-1)] <- 0
%o A004760 }
%o A004760 a <- which(b01 == 1)
%o A004760 # _Yosu Yurramendi_, Mar 30 2017
%o A004760 (Python)
%o A004760 def A004760(n): return m+(1<<m.bit_length()) if (m:=n-2)>0 else n-1 # _Chai Wah Wu_, Jul 26 2023
%Y A004760 Cf. A004761, A007814, A010060, A159559, A159560, A159615, A159619, A159629, A159698, A004759.
%K A004760 nonn,easy,base
%O A004760 1,3
%A A004760 _N. J. A. Sloane_
%E A004760 Offset changed to 1, b-file corrected. - _N. J. A. Sloane_, Aug 07 2016