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A003726 Numbers with no 3 adjacent 1's in binary expansion.

%I A003726 #47 Sep 04 2022 12:45:44
%S A003726 0,1,2,3,4,5,6,8,9,10,11,12,13,16,17,18,19,20,21,22,24,25,26,27,32,33,
%T A003726 34,35,36,37,38,40,41,42,43,44,45,48,49,50,51,52,53,54,64,65,66,67,68,
%U A003726 69,70,72,73,74,75,76,77,80,81,82
%N A003726 Numbers with no 3 adjacent 1's in binary expansion.
%C A003726 Positions of zeros in A014082. Could be called "tribbinary numbers" by analogy with A003714. - _John Keith_, Mar 07 2022
%C A003726 The sequence of Tribbinary numbers can be constructed by writing out the Tribonacci representations of nonnegative integers and then evaluating the result in binary. These numbers are similar to Fibbinary numbers A003714, Fibternary numbers A003726, and Tribternary numbers A356823. The number of Tribbinary numbers less than any power of two is a Tribonacci number. We can generate Tribbinary numbers recursively: Start by adding 0 and 1 to the sequence. Then, if x is a number in the sequence add 2x, 4x+1, and 8x+3 to the sequence. The n-th Tribbinary number is even if the n-th term of the Tribonacci word is a. Respectively, the n-th Tribbinary number is of the form 4x+1 if the n-th term of the Tribonacci word is b, and the n-th Tribbinary number is of the form 8x+3 if the n-th term of the Tribonacci word is c. Every nonnegative integer can be written as the sum of two Tribbinary numbers. Every number has a Tribbinary multiple. - _Tanya Khovanova_ and PRIMES STEP Senior, Aug 30 2022
%H A003726 Reinhard Zumkeller, <a href="/A003726/b003726.txt">Table of n, a(n) for n = 1..10000</a>
%H A003726 Robert Baillie and Thomas Schmelzer, <a href="https://library.wolfram.com/infocenter/MathSource/7166/">Summing Kempner's Curious (Slowly-Convergent) Series</a>, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
%H A003726 <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%F A003726 There are A000073(n+3) terms of this sequence with at most n bits. In particular, a(A000073(n+3)+1) = 2^n. - _Charles R Greathouse IV_, Oct 22 2021
%F A003726 Sum_{n>=2} 1/a(n) = 9.516857810319139410424631558212354346868048230248717360943194590798113163384... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - _Amiram Eldar_, Feb 13 2022
%t A003726 Select[Range[0, 82], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] (* _Michael De Vlieger_, Dec 23 2019 *)
%o A003726 (Haskell)
%o A003726 a003726 n = a003726_list !! (n - 1)
%o A003726 a003726_list = filter f [0..] where
%o A003726    f x = x < 7 || (x `mod` 8) < 7 && f (x `div` 2)
%o A003726 -- _Reinhard Zumkeller_, Jun 03 2012
%o A003726 (PARI) is(n)=!bitand(bitand(n, n<<1), n<<2) \\ _Charles R Greathouse IV_, Feb 11 2017
%Y A003726 Cf. A278038 (binary), A063037, A000073, A014082 (number of 111).
%Y A003726 Cf. A004781 (complement).
%Y A003726 Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110).
%K A003726 nonn,base,easy
%O A003726 1,3
%A A003726 _N. J. A. Sloane_