A261195 - OEIS

%I #37 Sep 14 2015 13:44:29

%S 0,1,6,7,16,17,22,23,40,41,46,47,56,57,62,63,384,385,390,391,400,401,

%T 406,407,424,425,430,431,440,441,446,447,576,577,582,583,592,593,598,

%U 599,616,617,622,623,632,633,638,639,960,961,966,967,976,977,982,983

%N Encoded symmetrical square binary matrices.

%C 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.

%H Philippe Beaudoin, <a href="/A261195/b261195.txt">Table of n, a(n) for n = 0..10000</a>

%F 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

%e 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:

%e   1 1 0

%e   1 0 1

%e   0 1 0

%e and the 4 X 4 matrix:

%e   1 1 0 0

%e   1 0 1 0

%e   0 1 0 0

%e   0 0 0 0

%e and all larger square matrices constructed in the same way. Since 391 is in the sequence, all these matrices are symmetrical.

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

%Y Cf. A231428, A261194.

%K nonn

%O 0,3

%A _Philippe Beaudoin_, Aug 11 2015