A261195 - OEIS

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

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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