A002820 Number of n X n invertible binary matrices A such that A+I is invertible. (Formerly M2170 N0866)

0, 2, 48, 5824, 2887680, 5821595648, 47317927329792, 1544457148312846336, 202039706313624586813440, 105823549214125066767168438272, 221819704567105547916502447159246848

Offset

1,2

Comments

Also number of linear orthomorphisms of GF(2)^n.

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..55

Zong Duo Dai, Solomon W. Golomb, and Guang Gong, Generating all linear orthomorphisms without repetition, Discrete Math. 205 (1999), 47-55.

P. F. Duvall, Jr. and P. W. Harley, III, A note on counting matrices, SIAM J. Appl. Math., 20 (1971), 374-377.

Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

Kent Morrison, Matrices over F_q with no eigenvalues of 0 or 1

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Index entries for sequences related to binary matrices

Formula

Reference gives a recurrence.

a(n) = 2^(n(n-1)/2) * A005327(n+1).

Maple

(Maple program based on Dai et al. from N. J. A. Sloane, Aug 10 2011)
N:=proc(n, i) option remember; if i = 1 then 1 else (2^n-2^(i-1))*N(n, i-1); fi; end;
Oh:=proc(n) option remember; local r; global N;
if n=0 then RETURN(1) elif n=1 then RETURN(0) else
add( 2^(r-2)*N(n, r)*2^(r*(n-r))*Oh(n-r), r=2..n); fi; end;
[seq(Oh(n), n=1..15)];

Mathematica

ni[n_, i_] := ni[n, i] = If[i == 1, 1, (2^n - 2^(i-1))*ni[n, i-1]]; a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[ 2^(r-2)*ni[n, r]*2^(r*(n-r))*a[n-r], {r, 2, n}]; Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Jan 19 2012, after Maple *)

Crossrefs

Cf. A002884.

Sequence in context: A067626 A053071 A238838 * A196448 A053290 A056989

Adjacent sequences: A002817 A002818 A002819 * A002821 A002822 A002823

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Mar 17 2000

Entry revised by N. J. A. Sloane, Aug 10 2011

Status

approved