A183159 - OEIS

%I #29 Apr 14 2024 03:47:12

%S 1,1,4,12,38,90,220,460,1018,2022,4304,8376,17566,33922,70756,136260,

%T 283682,545790,1135576,2184112,4543366,8737626,18174764,34951932,

%U 72700618,139809430,290804320,559239720,1163219438

%N 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).

%H R. Kehinde and A. Umar, <a href="http://arxiv.org/abs/1101.2558">On the semigroup of partial isometries of a finite chain</a>, arXiv:1101.2558 [math.GR], 2011.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-11,12,-4).

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

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

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

%F 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

%F a(n) = (51*2^n+(-2)^n-40)/12-n*(n+3). - _Jean-François Alcover_, Nov 22 2017

%e 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.

%p 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:

%p seq(A183159(n),n=0..50) ; # _R. J. Mathar_, Jan 06 2011

%t LinearRecurrence[{3,1,-11,12,-4},{1,1,4,12,38},30] (* _Harvey P. Dale_, Dec 06 2015 *)

%t a[n_] := (51*2^n+(-2)^n-40)/12-n*(n+3); Array[a, 29, 0] (* _Jean-François Alcover_, Nov 22 2017 *)

%Y Cf. A183158.

%K nonn

%O 0,3

%A _Abdullahi Umar_, Dec 28 2010