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A239304 Triangle of permutations corresponding to the compressed square roots of Gray code * bit-reversal permutation (A239303). 2
1, 1, 2, 3, 1, 2, 4, 2, 1, 3, 2, 5, 4, 1, 3, 2, 5, 6, 3, 1, 4, 6, 2, 3, 7, 5, 1, 4, 7, 3, 2, 6, 8, 4, 1, 5, 3, 8, 7, 2, 4, 9, 6, 1, 5, 3, 8, 9, 4, 2, 7, 10, 5, 1, 6, 9, 3, 4, 10, 8, 2, 5, 11, 7, 1, 6, 10, 4, 3, 9 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The symmetrical binary matrices corresponding to the rows of A239303 can be interpreted as adjacency matrices of undirected graphs. These graphs are chains where one end is connected to itself, so they can be interpreted as permutations. The end connected to itself is always the first element of the permutation, i.e., on the left side of the triangle.

Columns of the square array:

T(m,1) = A008619(m) = 1,2,2,3,3...

T(m,2) = 1,1,1...

T(m,3) = A028242(m+3) = 3,2,4,3,5,4,6,5,7,6,8,7,9,8,10,9,11,10,12...

T(m,4) = m+3 = 4,5,6...

T(m,5) = A084964(m+4) = 2,5,3,6,4,7,5,8,6,9,7,10,8,11,9,12,10,13...

T(m,6) = 2,2,2...

T(m,7) = A168230(m+5) = 6,3,7,4,8,5,9,6,10,7,11,8,12,9,13,10,14...

T(m,8) = m+6 = 7,8,9...

T(m,9) = A152832(m+9) = 3,8,4,9,5,10,6,11,7,12,8,13,9,14,10,15...

T(m,10) = 3,3,3...

Diagonals of the square array:

T(n,n) = a(A001844(n)) = 1,1,4,7,4,2,9,14,7,3,14,21,10,4,19,28,13,5,24...

T(n,2n-1) = a(A064225(n)) = 1,2,3...

T(2n-1,n) = a(A081267(n)) = 1,1,5,10,6,2,12,21,11,3,19,32,16,4,26,43,21...

LINKS

Tilman Piesk, First 140 rows of the triangle, flattened

Tilman Piesk, Sequency ordered Walsh matrix (Wikiversity)

Tilman Piesk, Calculation in MATLAB

EXAMPLE

Triangular array begins:

  1

  1 2

  3 1 2

  4 2 1 3

  2 5 4 1 3

  2 5 6 3 1 4

Square array begins:

  1 1 3 4 2 2

  2 1 2 5 5 2

  2 1 4 6 3 2

  3 1 3 7 6 2

  3 1 5 8 4 2

  4 1 4 9 7 2

Row 5 of A239303 is the vector (12,18,1,17,10), which corresponds to the following binary matrix:

  0 0 1 1 0

  0 1 0 0 1

  1 0 0 0 0

  1 0 0 0 1

  0 1 0 1 0

Interpreted as an adjacency matrix it describes the following graph, where each number is connected to its neighbors, and only the 2 is connected to itself:

  2 5 4 1 3

This is row 5 of the triangular array.

CROSSREFS

Cf. A239303, A028242, A084964, A168230, A152832.

Sequence in context: A026793 A344089 A329631 * A072193 A233359 A279345

Adjacent sequences:  A239301 A239302 A239303 * A239305 A239306 A239307

KEYWORD

nonn,tabl

AUTHOR

Tilman Piesk, Mar 14 2014

STATUS

approved

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Last modified August 10 10:09 EDT 2022. Contains 356039 sequences. (Running on oeis4.)