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A002820
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Number of n X n invertible binary matrices A such that A+I is invertible.
(Formerly M2170 N0866)
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5
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0, 2, 48, 5824, 2887680, 5821595648, 47317927329792, 1544457148312846336, 202039706313624586813440, 105823549214125066767168438272, 221819704567105547916502447159246848
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also number of linear orthomorphisms of GF(2)^n.
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REFERENCES
| Dai, Zong Duo; Golomb, Solomon W.; and Gong, Guang, 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.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
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).
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LINKS
| Kent Morrison, Matrices over F_q with no eigenvalues of 0 or 1
Index entries for sequences related to binary matrices
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FORMULA
| Reference gives a recurrence.
2^[n(n-1)/2] * A005327(n+1).
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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)];
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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}] (* From Jean-François Alcover, Jan 19 2012, after Maple *)
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CROSSREFS
| Cf. A002884.
Sequence in context: A087085 A067626 A053071 * A196448 A053290 A056989
Adjacent sequences: A002817 A002818 A002819 * A002821 A002822 A002823
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 17 2000
Entry revised by N. J. A. Sloane, Aug 10 2011
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