STATUS
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approved
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editing
approved
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 between elements x,y their binary concatenation between elements x,y.
proposed
editing
editing
proposed
(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
Empirically, there are A000010(n) positive terms with n binary digits. - Rémy Sigrist, Jan 01 2024
Rémy Sigrist, <a href="/A367745/a367745.png">Binary plot of the terms < 2^64</a>
proposed
editing
editing
proposed
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 between elements x,y their binary concatenation.
Luc Rousseau, <a href="/A367745/a367745_1.svg">SVG illustration of the nesting of the L(k) lists for n=1..9</a>.
Length of L(k) = 2^(k-1) + 1 = A000051(k-1).
(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
Luc Rousseau, <a href="/A367745/a367745.svg">SVG illustration of the nesting of the L(k) lists for n=1..9</a>.
Luc Rousseau, <a href="/A367745/a367745_1.svg">SVG illustration of the nesting of the L(k) lists for n=1..9</a>.
proposed
editing