A244961 Decimal equivalent of the binary string generated by the n X n antidiagonal matrix read by rows.

1, 6, 84, 4680, 1118480, 1108378656, 4467856773184, 72624976668147840, 4740885567116192841984, 1240362622532514091484054016, 1299343104384081987198247370753024, 5447177625434934817180194985991294945280

Offset

1,2

Comments

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).)

Links

Nathan J. Russell, Table of n, a(n) for n = 1..30

Formula

a(n) = Sum_{k=1..n} 2^((n-1)k).

a(1) = 1, a(n) = (2^(n^2)-2^n)/(2^n-2) when n>1.

Example

n=1 -> a(1) = 1 -> M = {{1}} -> M^2 = I_1 = 1.
n=2 -> a(2) = 6 = 0110_2 ->  M ={{0, 1}, {1, 0}} -> M^2 = I_2.
n=3 -> a(3) = 84 = 001010100_2 ->  M ={{0, 0, 1}, {0, 1, 0}, {1, 0, 0}}.

Mathematica

For[b = 2, b <= 10, ++b, Print[(2^b^2 - 2^b)/(2^b - 2)]; ]

Prog

(PARI) a(n) = if (n==1, 1, (2^(n^2)-2^n)/(2^n-2)); \\ Michel Marcus, Jul 11 2014
(Magma) [1] cat[(2^(n^2)-2^n)/(2^n-2): n in [2..15]]; // Vincenzo Librandi, Jul 26 2014

Crossrefs

Sequence in context: A365354 A195245 A030031 * A293455 A334516 A331014

Adjacent sequences: A244958 A244959 A244960 * A244962 A244963 A244964

Keyword

nonn,base

Author

Nathan J. Russell, Jul 08 2014

Extensions

Edited by Wolfdieter Lang, Jul 25 2014

Status

approved