OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..448
Wikipedia, Insertion sort
FORMULA
a(n) = a(n-1)*(n) + n! *(n+1)/2 - (n-1)!.
a(n) = n!/4 *(n^2+3*n-4*H(n)), where H(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Sep 02 2010
a(n) = ((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)) for n>2. - Alois P. Heinz, Dec 16 2016
a(n) = 2 * A285231(n+1). - Alois P. Heinz, Apr 15 2017
EXAMPLE
For n=3, insertion sorting 123, 213, 213, 231, 312, 321 takes 3+3+3+2+3+2 = 4*3+2*2 = 16 comparisons.
MAPLE
a:= proc(n) option remember;
`if`(n<2, 0, a(n-1)*n + (n-1)! * (n-1)*(n+2)/2)
end:
seq(a(n), n=0..30); # Alois P. Heinz, May 14 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 2][n+1],
((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Dec 16 2016
MATHEMATICA
a[n_] := n! ((n+1)(n+2)/4 - HarmonicNumber[n] - 1/2); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 12 2017, after Gary Detlefs *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Harmen Wassenaar (towr(AT)ai.rug.nl), Apr 10 2009
STATUS
approved