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 A001196 Double-bitters: only even length runs in binary expansion. 23

%I

%S 0,3,12,15,48,51,60,63,192,195,204,207,240,243,252,255,768,771,780,

%T 783,816,819,828,831,960,963,972,975,1008,1011,1020,1023,3072,3075,

%U 3084,3087,3120,3123,3132,3135,3264,3267,3276,3279,3312

%N Double-bitters: only even length runs in binary expansion.

%C Numbers whose set of base 4 digits is {0,3}. - _Ray Chandler_, Aug 03 2004

%C n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 4 for every i - _Ray Chandler_, Aug 03 2004

%C The first 2^n terms of the sequence could be obtained using the Cantor-like process for the segment [0,4^n-1]. E.g., for n=1 we have [0,{1,2},3] such that numbers outside of braces are the first 2 terms of the sequence; for n=2 we have [0,{1,2},3,{4,5,6,7,8,9,10,11},12,{13,14},15] such that the numbers outside of braces are the first 4 terms of the sequence, etc. - _Vladimir Shevelev_, Dec 17 2012

%C From _Emeric Deutsch_, Jan 26 2018: (Start)

%C Also, the indices of the compositions having only even parts. For the definition of the index of a composition see A298644. For example, 195 is in the sequence since its binary form is 11000011 and the composition [2,4,2] has only even parts. 132 is not in the sequence since its binary form is 10000100 and the composition [1,4,1,2] has also odd parts.

%C The command c(n) from the Maple program yields the composition having index n. (End)

%H Sean A. Irvine, <a href="/A001196/b001196.txt">Table of n, a(n) for n = 0..10000</a>

%H R. Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>

%H R. Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(2n) = 4a(n), a(2n+1) = 4a(n) + 3.

%F a(n) = 3 * A000695(n).

%p Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 3350 do if type(product(1+c(n)[j], j = 1 .. nops(c(n))), odd) = true then A := `union`(A, {n}) else end if end do: A; # most of the Maple program is due to _W. Edwin Clark_. - _Emeric Deutsch_, Jan 26 2018

%t fQ[n_] := Union@ Mod[Length@# & /@ Split@ IntegerDigits[n, 2], 2] == {0}; Select[ Range@ 10000, fQ] (* Or *)

%t fQ[n_] := Union@ Join[IntegerDigits[n, 4], {0, 3}] == {0, 3}; Select[ Range@ 10000, fQ] (* _Robert G. Wilson v_, Dec 24 2012 *)

%o a001196 n = if n == 0 then 0 else 4 * a001196 n' + 3 * b

%o where (n',b) = divMod n 2

%o -- _Reinhard Zumkeller_, Feb 21 2014

%Y 3 times the Moser-de Bruijn sequence A000695.

%Y Cf. A005823, A097252-A097262.

%K nonn,base,easy

%O 0,2

%A _N. J. A. Sloane_, based on an email from Bart la Bastide (bart(AT)xs4all.nl)

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Last modified January 21 11:00 EST 2019. Contains 319351 sequences. (Running on oeis4.)