%I #34 Sep 08 2022 08:46:08
%S 1,6,84,4680,1118480,1108378656,4467856773184,72624976668147840,
%T 4740885567116192841984,1240362622532514091484054016,
%U 1299343104384081987198247370753024,5447177625434934817180194985991294945280
%N Decimal equivalent of the binary string generated by the n X n antidiagonal matrix read by rows.
%C The binary representation of a(n), padded with zeros on the left to make an n^2-bit binary number, where the digits are grouped into n rows of n digits, form an n X n binary matrix whose entries along the antidiagonal are 1's and 0's everywhere else. a(n) will be the smallest integer whose binary representation forms an n X n involutory matrix. (I.e., let M be the n X n matrix formed by this binary representation of a(n); then a(n) is the smallest integer such that M^2 = I_n (the n X n unit matrix).)
%H Nathan J. Russell, <a href="/A244961/b244961.txt">Table of n, a(n) for n = 1..30</a>
%F a(n) = Sum_{k=1..n} 2^((n-1)k).
%F a(1) = 1, a(n) = (2^(n^2)-2^n)/(2^n-2) when n>1.
%e n=1 -> a(1) = 1 -> M = {{1}} -> M^2 = I_1 = 1.
%e n=2 -> a(2) = 6 = 0110_2 -> M ={{0, 1}, {1, 0}} -> M^2 = I_2.
%e n=3 -> a(3) = 84 = 001010100_2 -> M ={{0, 0, 1}, {0, 1, 0}, {1, 0, 0}}.
%t For[b = 2, b <= 10, ++b, Print[(2^b^2 - 2^b)/(2^b - 2)];]
%o (PARI) a(n) = if (n==1, 1, (2^(n^2)-2^n)/(2^n-2)); \\ _Michel Marcus_, Jul 11 2014
%o (Magma) [1] cat[(2^(n^2)-2^n)/(2^n-2): n in [2..15]]; // _Vincenzo Librandi_, Jul 26 2014
%K nonn,base
%O 1,2
%A _Nathan J. Russell_, Jul 08 2014
%E Edited by _Wolfdieter Lang_, Jul 25 2014