A252867 a(n) = n if n <= 2, otherwise the smallest number not occurring earlier having in its binary representation at least one bit in common with a(n-2), but none with a(n-1).

0, 1, 2, 5, 10, 4, 3, 12, 17, 6, 9, 18, 8, 7, 24, 33, 14, 32, 11, 36, 19, 40, 16, 13, 48, 15, 80, 34, 20, 35, 28, 65, 22, 41, 66, 21, 42, 68, 25, 38, 72, 23, 64, 26, 69, 50, 73, 52, 67, 44, 81, 46, 129, 30, 97, 130, 29, 98, 132, 27, 100, 131, 56, 70, 49, 74, 37, 82

Offset

0,3

Comments

Conjectured to be a permutation of the nonnegative integers. [Comment modified by N. J. A. Sloane, Jan 10 2015]

This is a purely set-based version of A098550, using the binary representation of numbers.

An equivalent definition in terms of sets: S(0) = {}, S(1) = {0}, S(2} = {1}; thereafter S(n) is the smallest set (different from the S{i} with i < n) of positive integers such that S(n) meets S(n-2) but is disjoint from S(n-1). - N. J. A. Sloane, Mar 27 2022; corrected Aug 01 2022.

Links

Chai Wah Wu, Table of n, a(n) for n = 0..50002 (First 10000 terms from Reinhard Zumkeller)

David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Chai Wah Wu, Scatterplot of first million terms

Chai Wah Wu, Scatterplot of first million terms, with red lines powers of 2.

Chai Wah Wu, Gzipped file with first million terms [Save file, delete .txt suffix, then open]

Example

The sequence of sets is {}, {0}, {1}, {0,2}, {1,3}, {2}, {0,1}, {2,3}. After the initial 3 terms, S(n) is the minimum set (as ordered by A048793) that has a nonempty intersection with S(n-2) but empty intersection with S(n-1). [Typos corrected by N. J. A. Sloane, Aug 01 2022 at the suggestion of Michel Dekking.]
Comment from N. J. A. Sloane, Dec 31 2014: The binary expansions of the first few terms are:
0  = 000000
1  = 000001
2  = 000010
5  = 000101
10 = 001010
4  = 000100
3  = 000011
12 = 001100
17 = 010001
6  = 000110
9  = 001001
18 = 010010
8  = 001000
7  = 000111
24 = 011000
33 = 100001
14 = 001110
32 = 100000
11 = 001011
36 = 100100
19 = 010011
40 = 101000
...

Mathematica

a[n_] := a[n] = If[n<3, n, For[k=3, True, k++, If[FreeQ[Array[a, n-1], k], If[BitAnd[k, a[n-2]] >= 1 && BitAnd[k, a[n-1]] == 0, Return[k]]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 03 2018 *)

Prog

(PARI) invecn(v, k, x)=for(i=1, k, if(v[i]==x, return(i))); 0
alist(n)=local(v=vector(n, i, i-1), x); for(k=4, n, x=3; while(invecn(v, k-1, x)||!bitand(v[k-2], x)||bitand(v[k-1], x), x++); v[k]=x); v
(Haskell)
import Data.Bits ((.&.)); import Data.List (delete)
a252867 n = a252867_list !! n
a252867_list = 0 : 1 : 2 : f 1 2 [3..] where
   f :: Int -> Int -> [Int] -> [Int]
   f u v ws = g ws where
     g (x:xs) = if x .&. u > 0 && x .&. v == 0
                   then x : f v x (delete x ws) else g xs
-- Reinhard Zumkeller, Dec 24 2014
(Python)
A252867_list, l1, l2, s, b = [0, 1, 2], 2, 1, 3, set()
for _ in range(10**2):
    i = s
    while True:
        if not (i in b or i & l1) and i & l2:
            A252867_list.append(i)
            l2, l1 = l1, i
            b.add(i)
            while s in b:
                b.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 27 2014

Crossrefs

Cf. A098550, A252865, A048793, A252868.

Reading this sequence mod 2 gives A253050 and A253051.

Cf. A253581, A253582, A253589 (binary weight), A253603.

Analyzed further in A303596, A303597, A303598, A303599, A305368.

The graphs of A109812, A252867, A305369, A305372 all have roughly the same, mysterious, fractal-like structure. - N. J. A. Sloane, Jun 03 2018

Sequence in context: A064365 A177356 A078322 * A352199 A194356 A227317

Adjacent sequences: A252864 A252865 A252866 * A252868 A252869 A252870

Keyword

nonn,base

Author

Franklin T. Adams-Watters, Dec 23 2014

Status

approved