A353709 a(0)=0, a(1)=1; thereafter a(n) = smallest nonnegative integer not among the earlier terms of the sequence such that a(n) and a(n-2) have no common 1-bits in their binary representations and also a(n) and a(n-1) have no common 1-bits in their binary representations.

0, 1, 2, 4, 8, 3, 16, 12, 32, 17, 6, 40, 64, 5, 10, 48, 65, 14, 128, 33, 18, 68, 9, 34, 20, 72, 35, 132, 24, 66, 36, 25, 130, 96, 13, 144, 98, 256, 21, 42, 192, 257, 22, 104, 129, 258, 28, 97, 384, 26, 37, 320, 136, 7, 80, 160, 11, 84, 288, 131, 76, 272, 161, 70, 264, 49, 134, 328, 512, 19, 44, 448, 513, 30, 224, 768, 15, 112, 640, 259, 52, 200, 514, 53

Offset: 0

Author

N. J. A. Sloane, May 06 2022

Keywords

• nonn
• base

Comments

A set-theory analog of A084937.

Conjecture: This is a permutation of the nonnegative numbers.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384
• Walter Trump, Table of n, a(n) for n = 0..10^6
• Walter Trump, Log-log plot of first 2^24 terms

Crossrefs

Cf. A084937 (number theory analog), A109812, A121216, A353405 (powers of 2), A353708, A353710, A353715 and A353716 (a(n)+a(n+1)), A353717 (inverse), A353718, A353719 (primes), A353720 and A353721 (Records).

For the numbers that are the slowest to appear see A353723 and A353722.

Programs

Maple

read(transforms) : # ANDnos def'd here
A353709 := proc(n)
option remember;
local c, i, known ;
if n <= 2 then
n;
else
for c from 1 do
known := false ;
for i from 1 to n-1 do
if procname(i) = c then
known := true;
break ;
end if;
end do:
if not known and ANDnos(c, procname(n-2)) = 0 and ANDnos(c, procname(n-1)) = 0 then
return c;
end if;
end do:
end if;
end proc: # Following R. J. Mathar's program for A109812.
[seq(A353709(n), n=0..256)] ;
# second Maple program:
b:= proc() false end: t:= 2:
a:= proc(n) option remember; global t; local k; if n<2 then n
      else for k from t while b(k) or Bits[And](k, a(n-2))>0
      or Bits[And](k, a(n-1))>0 do od; b(k):=true;
      while b(t) do t:=t+1 od; k fi
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 06 2022

Mathematica

nn = 83; c[] = -1; a[0] = c[0] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[a[n - 1], k] == 0, BitAnd[a[n - 2], k] == 0], k++]; Set[{a[n], c[k]}, {k, n}]; If[k == u, While[c[u] > -1, u++]], {n, 2, nn}]; Array[a, nn+1, 0] (* _Michael De Vlieger, May 06 2022 *)

Python

from itertools import count, islice
def A353709_gen(): # generator of terms
    s, a, b, c, ab = {0,1}, 0, 1, 2, 1
    yield from (0,1)
    while True:
        for n in count(c):
            if not (n & ab or n in s):
                yield n
                a, b = b, n
                ab = a|b
                s.add(n)
                while c in s:
                    c += 1
                break
A353709_list = list(islice(A353709_gen(),20)) # Chai Wah Wu, May 07 2022