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%I #47 Sep 04 2022 12:45:44
%S 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 34,35,36,37,38,40,41,42,43,44,45,48,49,50,51,52,53,54,64,65,66,67,68,
%U 69,70,72,73,74,75,76,77,80,81,82
%N Numbers with no 3 adjacent 1's in binary expansion.
%C Positions of zeros in A014082. Could be called "tribbinary numbers" by analogy with A003714. - _John Keith_, Mar 07 2022
%C 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 Reinhard Zumkeller, <a href="/A003726/b003726.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%F 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 Sum_{n>=2} 1/a(n) = 9.516857810319139410424631558212354346868048230248717360943194590798113163384... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - _Amiram Eldar_, Feb 13 2022
%t Select[Range[0, 82], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] (* _Michael De Vlieger_, Dec 23 2019 *)
%o (Haskell)
%o a003726 n = a003726_list !! (n - 1)
%o a003726_list = filter f [0..] where
%o f x = x < 7 || (x `mod` 8) < 7 && f (x `div` 2)
%o -- _Reinhard Zumkeller_, Jun 03 2012
%o (PARI) is(n)=!bitand(bitand(n, n<<1), n<<2) \\ _Charles R Greathouse IV_, Feb 11 2017
%Y Cf. A278038 (binary), A063037, A000073, A014082 (number of 111).
%Y Cf. A004781 (complement).
%Y Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110).
%K nonn,base,easy
%O 1,3
%A _N. J. A. Sloane_
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