A183159 The number of partial isometries (of an n-chain) of fix zero (fix of alpha = 0). Equivalently, the number of partial derangement isometries (of an n-chain).

1, 1, 4, 12, 38, 90, 220, 460, 1018, 2022, 4304, 8376, 17566, 33922, 70756, 136260, 283682, 545790, 1135576, 2184112, 4543366, 8737626, 18174764, 34951932, 72700618, 139809430, 290804320, 559239720, 1163219438

Offset: 0

Author

Abdullahi Umar, Dec 28 2010

Keywords

• nonn

Examples

			a(2) = 4 because there are exactly 4 partial derangement isometries (on a 2-chain) , namely: empty map; 1-->2; 2-->1; (1,2)-->(2,1). a(3) = 12 because there are exactly 12 partial isometries (on a 3-chain) namely: empty map; 1-->2; 1-->3; 2-->1; 2-->3; 3-->1; 3-->2; (1,2)-->(2,1); (1,2)-->(2,3); (2,3)-->(1,2); (2,3)-->(3,2); (1,3)-->(3,1) - the mappings are coordinate-wise.
		

Links

• R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.
• Index entries for linear recurrences with constant coefficients, signature (3,1,-11,12,-4).

Crossrefs

Cf. A183158.

Programs

Maple

A183159 := proc(n) nh := floor(n/2) ; if type(n,'even') then 13*4^nh-12*nh^2-18*nh-10; else 25*4^nh-12*nh^2-30*nh-22; end if; %/3 ; end proc:
seq(A183159(n),n=0..50) ; # R. J. Mathar, Jan 06 2011

Mathematica

LinearRecurrence[{3,1,-11,12,-4},{1,1,4,12,38},30] (* Harvey P. Dale, Dec 06 2015 *)
a[n_] := (51*2^n+(-2)^n-40)/12-n*(n+3); Array[a, 29, 0] (* Jean-François Alcover, Nov 22 2017 *)

Formula

a(2n) = (13*4^n-12*n^2-18*n-10)/3, n>=0.

a(2n+1) = (25*4^n-12*n^2-30*n-22)/3, n>=0.

a(n) = A183158(n,0).

G.f.: ( 1-2*x-3*x^4+10*x^3 ) / ( (2*x-1)*(2*x+1)*(x-1)^3 ). - Joerg Arndt, Dec 30 2010

a(n) = (51*2^n+(-2)^n-40)/12-n*(n+3). - Jean-François Alcover, Nov 22 2017