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
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
Programs
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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]
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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
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