cp's OEIS Frontend

A357082 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of a(n-1) + a(n) does not appear in the binary string concatenation of a(0)..a(n-1).

%I A357082 #35 Feb 15 2023 07:49:10
%S A357082 0,1,2,3,4,5,10,6,9,7,17,12,20,13,16,18,11,21,15,19,23,38,26,8,24,37,
%T A357082 27,34,30,31,33,32,40,36,28,44,41,35,29,43,39,25,47,53,51,49,42,22,50,
%U A357082 14,58,46,54,62,65,63,66,67,61,68,60,69,59,70,74,55,73,56,72,57,71,75,78,76,52,77,92
%N A357082 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of a(n-1) + a(n) does not appear in the binary string concatenation of a(0)..a(n-1).
%C A357082 The main concentration of terms lie along the line a(n) = n, so numerous fixed points exists - there are 301 fixed points in the first 300000 terms. The sequence is conjectured to be a permutation of the positive integers.
%H A357082 Scott R. Shannon, <a href="/A357082/b357082.txt">Table of n, a(n) for n = 0..10000</a>
%H A357082 Michael De Vlieger, <a href="/A357082/a357082_3.png">Scalar plot of a(n)</a>, n = 0..2^16, highlighting records in red, local minima in blue, a(n) = 2^e in green, labeling the indices of certain records and local minima.
%H A357082 Michael De Vlieger, <a href="/A357082/a357082_4.png">Bitmap of a(n)</a>, n = 0..2^11, 12X vertical exaggeration, read horizontally where black represents 1 and white 0, with least significant bit on bottom.
%H A357082 Michael De Vlieger, <a href="/A357082/a357082_5.png">Bitmap of (a(n-1) + a(n))</a>, n = 1..2^11, 12X vertical exaggeration, read horizontally where black represents 1 and white 0, with least significant bit on bottom.
%H A357082 Scott R. Shannon, <a href="/A357082/a357082.png">Image for n=0..20000</a>.
%H A357082 Scott R. Shannon, <a href="/A357082/a357082_1.png">Image for n=0..75000</a>.
%H A357082 Scott R. Shannon, <a href="/A357082/a357082_7.png">Image for n=0..300000</a>.
%H A357082 Scott R. Shannon, <a href="/A357082/a357082_8.png">Image for n=0..2250000</a>.
%e A357082 a(6) = 10 as the concatenation of a(0)..a(5) in binary is "011011100101" and a(5) + 10 = 5 + 10 = 15 = 1111_2 which does not appear in the concatenated string. Since  11 = 1011_2, 12 = 1100_2, 13 = 1101_2, 14 = 1110_2 all appear in the concatenated string, a(6) cannot be 6, 7, 8 or 9.
%t A357082 nn = 77; c[_] = False; j = a[0] = 0; u = 1; w = "0"; Do[k = u; While[Or[c[k], StringContainsQ[w, Set[v, IntegerString[j + k, 2]]]], k++]; Set[{a[n], c[k]}, {k, True}]; Set[{j, w}, {k, w <> IntegerString[k, 2]}]; If[k == u, While[c[u], u++]], {n, nn}]; Array[a, nn + 1, 0] (* _Michael De Vlieger_, Sep 13 2022 *)
%o A357082 (Python)
%o A357082 from itertools import islice
%o A357082 def agen():
%o A357082     aset, astr, an, mink = {0}, "0", 0, 1
%o A357082     while True:
%o A357082         yield an; k = mink
%o A357082         while k in aset or bin(an+k)[2:] in astr: k += 1
%o A357082         while mink in aset: mink += 1
%o A357082         an = k; aset.add(an); astr += bin(an)[2:]
%o A357082 print(list(islice(agen(), 77))) # _Michael S. Branicky_, Sep 16 2022
%Y A357082 Cf. A007088, A030302, A118248, A341766.
%K A357082 nonn,base,look
%O A357082 0,3
%A A357082 _Scott R. Shannon_, Sep 11 2022
%E A357082 Images for n=0..300000 and n=0..2250000 corrected by _Scott R. Shannon_, Oct 05 2022