OFFSET
0,5
REFERENCES
T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 468.
Todd Simpson, Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.
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 = 0..200
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]. See Vol. 3 page 468. There may have been some confusion here of this sequence with A003471.
T. Simpson, Letter to N. J. A. Sloane, Mar. 1992.
T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)
FORMULA
a(n) = (n-1)*a(n-1) + 2*(n-d)*a(n-e), where (d, e) = (2, 3) if n even, (1, 2) if n odd.
Recurrence (for n>=7): (3*n^2 - 17*n + 23)*a(n) = (3*n^2 - 17*n + 21)*a(n-1) + (3*n^4 - 23*n^3 + 63*n^2 - 74*n + 34)*a(n-2) - 4*(n-3)*(n-2)*a(n-3) + 2*(n-4)*(n-3)*(3*n^2 - 11*n + 9)*a(n-4). - Vaclav Kotesovec, Mar 07 2014
a(n) ~ c * n!, where c = 5*sinh(sqrt(2))/2^(3/2) - 3*cosh(sqrt(2))/2 = 0.15347184510862040153106983922669125715345689997588202335369... - Vaclav Kotesovec, Mar 07 2014, updated Apr 20 2024
MAPLE
a:= proc(n) option remember; `if`(n<5, [1, 0$3, 4][n+1],
(n-1)*a(n-1)+2*`if`(n::even, (n-2)*a(n-3), (n-1)*a(n-2)))
end:
seq(a(n), n=0..23); # Alois P. Heinz, Jun 27 2020
MATHEMATICA
nmax = 20; b = ConstantArray[0, nmax+1]; b[[1]] = 1; b[[2]] = 0; b[[3]] = 0; b[[4]] = 0; b[[5]] = 4; Do[b[[n+1]] = (n-1)*b[[n]] + If[EvenQ[n], 2*(n-2)*b[[n-2]], 2*(n-1)*b[[n-1]]], {n, 5, nmax}]; b (* Vaclav Kotesovec, Mar 07 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Sep 24 2001
STATUS
approved