A261195 Encoded symmetrical square binary matrices.

0, 1, 6, 7, 16, 17, 22, 23, 40, 41, 46, 47, 56, 57, 62, 63, 384, 385, 390, 391, 400, 401, 406, 407, 424, 425, 430, 431, 440, 441, 446, 447, 576, 577, 582, 583, 592, 593, 598, 599, 616, 617, 622, 623, 632, 633, 638, 639, 960, 961, 966, 967, 976, 977, 982, 983

Offset

0,3

Comments

We encode an n X n binary matrix reading it antidiagonal by antidiagonal, starting from the least significant bit. A given entry in the sequence therefore represents the infinite family of n X n matrices that can be obtained by adding zero antidiagonals. All of these matrices are symmetrical. This encoding makes it possible to obtain a sequence rather than a table.

Links

Philippe Beaudoin, Table of n, a(n) for n = 0..10000

Formula

a((2n+1)*2^(k-1)) = a(n*2^k) + a(2^(k-1)) for n >= 0 and k >= 1. - Eric Werley, Sep 13 2015

Example

391 = 0b110000111 encodes all square matrices with the first four antidiagonals equal to ((1), (1, 1), (0, 0, 0), (0, 1, 1, 0)), for example, the 3 X 3 matrix:
  1 1 0
  1 0 1
  0 1 0
and the 4 X 4 matrix:
  1 1 0 0
  1 0 1 0
  0 1 0 0
  0 0 0 0
and all larger square matrices constructed in the same way. Since 391 is in the sequence, all these matrices are symmetrical.

Mathematica

b[n_] := Select[ Tuples[{0, 1}, n], # == Reverse@ # &]; FromDigits[#, 2]& /@ Join @@@ Tuples[ b/@ Range[7, 1, -1]] (* Giovanni Resta, Aug 12 2015 *)

Crossrefs

Cf. A231428, A261194.

Sequence in context: A078388 A344222 A315843 * A008538 A000870 A315844

Adjacent sequences: A261192 A261193 A261194 * A261196 A261197 A261198

Keyword

nonn

Author

Philippe Beaudoin, Aug 11 2015

Status

approved