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A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).

%I A048716 #35 Oct 28 2021 17:59:33
%S A048716 0,1,2,3,4,6,8,9,12,16,17,18,19,24,25,32,33,34,35,36,38,48,49,50,51,
%T A048716 64,65,66,67,68,70,72,73,76,96,97,98,99,100,102,128,129,130,131,132,
%U A048716 134,136,137,140,144,145,146,147,152,153,192,193,194,195,196,198,200,201
%N A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).
%C A048716 If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.
%C A048716 It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - _Benoit Cloitre_, Mar 09 2003
%C A048716 Yes, as remarked in A048715, "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." - _Jason Kimberley_, Dec 21 2011
%C A048716 A116361(a(n)) <= 2. - _Reinhard Zumkeller_, Feb 04 2006
%H A048716 Charles R Greathouse IV, <a href="/A048716/b048716.txt">Table of n, a(n) for n = 1..10000</a>
%H A048716 <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%H A048716 <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>
%H A048716 <a href="/index/Con#CongruXOR">Index entries for sequences defined by congruent products under XOR</a>
%t A048716 Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
%t A048716 (* Second program: *)
%t A048716 filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{___, 0, _, 1, ___}|{___ 1, _, 0, ___}] && !MatchQ[bb, {___, 1, _, 1, ___}]];
%t A048716 Select[Range[0, 201], filterQ] (* _Jean-François Alcover_, Dec 31 2020 *)
%o A048716 (PARI) is(n)=!bitand(n,n>>2) \\ _Charles R Greathouse IV_, Oct 03 2016
%o A048716 (PARI) list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<<valuation(t,2), listput(v,n); n++)); Vec(v) \\ _Charles R Greathouse IV_, Oct 22 2021
%Y A048716 Superset of A048715 and A048719. Union of A004742 and A003726.
%Y A048716 Cf. A048729, A003714, A115845, A115847, A116360.
%K A048716 nonn,base,easy
%O A048716 1,3
%A A048716 Antti Karttunen, Mar 30 1999