A165191 Irregular triangle B(n,i) = i-th significant bit of Gray code of n.

0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1

Offset

0

Comments

The n-th row has length A070939(n); a nondecreasing sequence.

Each row, when interpreted as a finite sequence can be mapped via Euler's Transform to familiar integer sequences.

Note that adjacent rows differ by only one "bit" which simplifies the transition from one row to the next. (cf. A003188, A055975 and A119972).

Links

Table of n, a(n) for n=0..104.

Formula

The n-th row is the reversed bit string of A003188(n). Namely, A003188(n) = Sum_{0<=i<A070939(n)} B(n,i) 2^i.

Example

The triangle begins: 0 1 11 01 011 111 101 001 0011 1011 1111 0111 0101 1101 1001 0001 00011 10011 11011 11111 ...

Maple

B:= proc(n) option remember; local b; b:= ilog2(n);
      `if`(n<=1, n, zip((x, y)->x+y, [B(2^(b+1)-1-n)], [0$b, 1], 0)[])
    end:
seq(B(n), n=0..30);  # Alois P. Heinz, May 21 2012

Mathematica

zip = With[{m = Max[Length[#1], Length[#2]]}, PadRight[#1, m] + PadRight[#2, m]]&; B[n_] := B[n] = With[{b = Floor[Log[2, n]]}, If[n <= 1, {n}, zip[B[2^(b+1)-1-n], Append[Array[0&, b], 1]]]]; Table[B[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Feb 13 2017, after Alois P. Heinz *)

Prog

(Magma) // Recursive
N := 5; s := [[]];
for n in [1..N] do
  for j in [#s..1 by -1] do
    Append(~s, Append(s[j], 1));
    Append(~s[j], 0);
  end for;
end for;
&cat[IntegerToSequence(SequenceToInteger(b, 2), 2):b in s];
(Magma) // Direct
B:=func<n|[(s[i]+s[i+1])mod 2:i in[1..#s-1]]cat[s[#s]]where s is IntegerToSequence(n, 2)>;

Crossrefs

Euler transforms of the rows begin: A000007, A000012, A008619, A059841, A103221, A001399, A008620, A022003, A008679, A025767, A001400.

Sequence in context: A168395 A278718 A130854 * A189576 A112448 A101264

Adjacent sequences: A165188 A165189 A165190 * A165192 A165193 A165194

Keyword

nonn,easy,tabf

Author

Alford Arnold, Sep 29 2009

Extensions

Edited by Jason Kimberley, Mar 31 2012

Status

approved