

A003726


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


28



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, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 80, 81, 82
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OFFSET

1,3


COMMENTS

Positions of zeros in A014082. Could be called "tribbinary numbers" by analogy with A003714.  John Keith, Mar 07 2022
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 nth Tribbinary number is even if the nth term of the Tribonacci word is a. Respectively, the nth Tribbinary number is of the form 4x+1 if the nth term of the Tribonacci word is b, and the nth Tribbinary number is of the form 8x+3 if the nth 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


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (SlowlyConvergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2automatic sequences.


FORMULA

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
Sum_{n>=2} 1/a(n) = 9.516857810319139410424631558212354346868048230248717360943194590798113163384... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links).  Amiram Eldar, Feb 13 2022


MATHEMATICA

Select[Range[0, 82], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] (* Michael De Vlieger, Dec 23 2019 *)


PROG

(Haskell)
a003726 n = a003726_list !! (n  1)
a003726_list = filter f [0..] where
f x = x < 7  (x `mod` 8) < 7 && f (x `div` 2)
 Reinhard Zumkeller, Jun 03 2012
(PARI) is(n)=!bitand(bitand(n, n<<1), n<<2) \\ Charles R Greathouse IV, Feb 11 2017


CROSSREFS

Cf. A278038 (binary), A063037, A000073, A014082 (number of 111).
Cf. A004781 (complement).
Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110).
Sequence in context: A325114 A004765 A247063 * A343110 A337582 A004828
Adjacent sequences: A003723 A003724 A003725 * A003727 A003728 A003729


KEYWORD

nonn,base,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



