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A001196 Double-bitters: only even length runs in binary expansion. 20
0, 3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255, 768, 771, 780, 783, 816, 819, 828, 831, 960, 963, 972, 975, 1008, 1011, 1020, 1023, 3072, 3075, 3084, 3087, 3120, 3123, 3132, 3135, 3264, 3267, 3276, 3279, 3312 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Numbers whose set of base 4 digits is {0,3}. - Ray Chandler, Aug 03 2004

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

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

From Emeric Deutsch, Jan 26 2018: (Start)

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.

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

LINKS

Sean A. Irvine, Table of n, a(n) for n = 0..10000

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Index entries for sequences related to binary expansion of n

FORMULA

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

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

MAPLE

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

MATHEMATICA

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

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

PROG

(Haskell)

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

            where (n', b) = divMod n 2

-- Reinhard Zumkeller, Feb 21 2014

CROSSREFS

3 times the Moser-de Bruijn sequence A000695.

Cf. A005823, A097252-A097262.

Sequence in context: A022380 A290593 A005392 * A096854 A013191 A009783

Adjacent sequences:  A001193 A001194 A001195 * A001197 A001198 A001199

KEYWORD

nonn,base,easy

AUTHOR

N. J. A. Sloane, based on an email from Bart la Bastide (bart(AT)xs4all.nl)

STATUS

approved

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Last modified June 22 17:19 EDT 2018. Contains 305672 sequences. (Running on oeis4.)