A367745 Numbers which occur anywhere in A367795, i.e. in lists L(k) where L(1) = [1,0] and L(k+1) is obtained from L(k) by inserting their binary concatenation between elements x,y.

0, 1, 2, 4, 6, 8, 14, 16, 20, 26, 30, 32, 62, 64, 72, 84, 106, 118, 126, 128, 164, 218, 254, 256, 272, 340, 426, 494, 510, 512, 584, 950, 1022, 1024, 1056, 1160, 1316, 1364, 1706, 1754, 1910, 2014, 2046, 2048, 2708, 3434, 4094, 4096, 4160, 4368, 4680, 5284, 5460, 6826, 7002, 7606, 7918, 8126, 8190

Offset: 1

Author

Luc Rousseau, Nov 29 2023

Keywords

• nonn
• base

Comments

0 is considered to be a 1-bit-long number and has 0 for binary expansion.

Empirically, there are A000010(n) positive terms with n binary digits. - Rémy Sigrist, Jan 01 2024

Examples

			The L(k) lists written in binary begin:
L(1) = [1, 0]
L(2) = [1, 10, 0] -- 10 inserted between 1 and 0
L(3) = [1, 110, 10, 100, 0] -- 110 inserted between 1 and 10, 100 between 10 and 0
L(4) = [1, 1110, 110, 11010, 10, 10100, 100, 1000, 0] -- etc.
0, 1, 10, 100, 110, 1000, ... are producible binary expansions, so the corresponding numbers (0, 1, 2, 4, 6, 8, ...) are in this sequence.
		

Links

• Rémy Sigrist, Binary plot of the terms < 2^64

Crossrefs

Cf. A000010, A367795 (the triangle of L(k) lists).

Programs

PARI

sz(n)=if(n==0,1,logint(n,2)+1)
L(n)=if(n==1, List([1, 0]), my(LL=L(n-1), k=#LL); while(k>1, listinsert(LL, (LL[k-1] << sz(LL[k])) + LL[k], k); k--); LL)
list_a(depth)=my(aa=vecsort(L(depth)), i=1, j=2^depth); while(i<=#aa&&aa[i]
				

PARI

explore(w, p, s) = { my (ps=concat(p, s)); if (#ps <= w, if (nb++ > #vv, vv=concat(vv, vector(#vv))); vv[nb]=fromdigits(ps,2); explore(w, p, ps); explore(w, ps, s);); }
list_a(w) = { nb = 2; vv = [1,0]; explore(w,[1],[0]); Set(vv[1..nb]); } \\ terms < 2^w; Rémy Sigrist, Jan 01 2024

Python

from itertools import chain, count, islice, zip_longest
def agen(): # generator of terms
    L = ["1", "0"]
    for k in count(1):
        yield from sorted(int(t, 2) for t in L if len(t) == k)
        Lnew = [s+t for s, t in zip(L[:-1], L[1:])]
        L = [t for t in chain(*zip_longest(L, Lnew)) if t is not None]
print(list(islice(agen(), 60))) # Michael S. Branicky, Nov 30 2023