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%I #61 Dec 31 2022 15:19:49
%S 0,1,2,4,8,9,16,17,18,32,33,34,36,64,65,66,68,72,73,128,129,130,132,
%T 136,137,144,145,146,256,257,258,260,264,265,272,273,274,288,289,290,
%U 292,512,513,514,516,520,521,528,529,530,544,545,546,548,576,577,578,580
%N Binary expansion matches (100(0)*)*(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).
%C No more than one 1-bit in each bit triple.
%C All terms satisfy A048727(n) = 7*n.
%C Constructed from A000930 in the same way as A003714 is constructed from A000045.
%C It appears that n is in the sequence if and only if C(7n,n) is odd (cf. A003714). - _Benoit Cloitre_, Mar 09 2003
%C The conjecture by Benoit is correct. This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p. - _Franklin T. Adams-Watters_, Oct 06 2009
%H G. C. Greubel, <a href="/A048715/b048715.txt">Table of n, a(n) for n = 0..1275</a>
%H Sebastian Karlsson, <a href="/A048715/a048715.txt">Walnut code that verifies the conjectures of Paul D. Hanna</a>
%H Walnut can be downloaded from <a href="https://cs.uwaterloo.ca/~shallit/walnut.html">https://cs.uwaterloo.ca/~shallit/walnut.html</a>.
%H <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>
%H <a href="/index/Con#CongruXOR">Index entries for sequences defined by congruent products under XOR</a>
%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%F a(0) = 0, a(n) = (2^(invfoo(n)-1))+a(n-foo(invfoo(n))), where foo(n) is foo(n-1) + foo(n-3) (A000930) and invfoo is its "integral" (floored down) inverse.
%F a(n) XOR 6*a(n) = 7*a(n); 3*a(n) XOR 4*a(n) = 7*a(n); 3*a(n) XOR 5*a(n) = 6*a(n); (conjectures). - _Paul D. Hanna_, Jan 22 2006
%F The conjectures can be verified using the Walnut theorem-prover (see links). - _Sebastian Karlsson_, Dec 31 2022
%t Reap[Do[If[OddQ[Binomial[7n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
%t (* Second program: *)
%t filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, !MatchQ[bb, {___, 1, 0, 1, ___}|{___, 1, 1, ___}]];
%t Select[Range[0, 580], filterQ] (* _Jean-François Alcover_, Dec 31 2020 *)
%o (PARI) is(n)=!bitand(n, 6*n) \\ _Charles R Greathouse IV_, Oct 03 2016
%o (Perl) for my $k (0..580) { print "$k, " if sprintf("%b", $k) =~ m{^(100(0)*)*(0|1|10)?$}; } # _Georg Fischer_, Jun 26 2021
%o (Python)
%o import re
%o def ok(n): return re.fullmatch('(100(0)*)*(0|1|10)?', bin(n)[2:]) != None
%o print(list(filter(ok, range(581)))) # _Michael S. Branicky_, Jun 26 2021
%Y Subsequence of A048716.
%Y Cf. A003726, A004742, A004743, A004744, A048717, A048718, A048719, A048730, A048733, A115422, A115423, A115424.
%K nonn,base,easy
%O 0,3
%A _Antti Karttunen_, Mar 30 1999
%E Definition corrected by _Georg Fischer_, Jun 26 2021
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