|
%I #45 Dec 31 2022 05:39:24
%S 1,2,7,22,59,142,319,686,1435,2950,5999,12118,24379,48926,98047,
%T 196318,392891,786070,1572463,3145286,6290971,12582382,25165247,
%U 50331022,100662619,201325862,402652399,805305526,1610611835,3221224510,6442449919,12884900798
%N The number T(n) of isometries of all subspaces of the finite metric space {1,2,...,n} (as a subspace of the reals with the Euclidean metric).
%C Also the number of (not necessarily maximal) cliques in the n X n bishop graph. - _Eric W. Weisstein_, Jun 04 2017
%H Vincenzo Librandi, <a href="/A183156/b183156.txt">Table of n, a(n) for n = 0..1000</a>
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F T(n) = 3*2^(n+1) - (n+2)^2 - 1, (n >= 0).
%F G.f.: (1 - 3*x + 6*x^2 - 2*x^3) / ( (2*x - 1)*(x - 1)^3 ). - _R. J. Mathar_, Jul 03 2011
%F a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4). - _Eric W. Weisstein_, Nov 29 2017
%F a(n) = A295909(n) + A295910(n) for n > 1. - _Eric W. Weisstein_, Nov 29 2017
%F a(n) = 2*a(n-1) + n^2 - 1. - _Kritsada Moomuang_, Oct 25 2019
%e T(2) = 7 because there are exactly 7 partial isometries (on a 2-chain), namely: empty map; 1-->1; 1-->2; 2-->1; 2-->2; (1,2)-->(1,2); (1,2)-->(2,1) - the mappings are coordinate-wise.
%t LinearRecurrence[{5, -9, 7, -2}, {1, 2, 7, 22}, 40] (* _Vincenzo Librandi_, Oct 11 2017 *)
%t Table[3 2^(n + 1) - (n + 2)^2 - 1, {n, 0, 30}] (* _Vincenzo Librandi_, Oct 11 2017 *)
%t LinearRecurrence[{5, -9, 7, -2}, {2, 7, 22, 59}, {0, 20}] (* _Eric W. Weisstein_, Nov 29 2017 *)
%t CoefficientList[Series[(1 - 3 x + 6 x^2 - 2 x^3)/((-1 + x)^3 (-1 + 2 x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Nov 29 2017 *)
%o (PARI) for(n=0,33,print1(3*(2^(n+1))-(n+2)^2-1,", "))
%o (Magma) [3*2^(n+1)-(n+2)^2-1: n in [0..33]]; // _Vincenzo Librandi_, Oct 11 2017
%Y Row sums of triangles A183157, A183158.
%Y Cf. A295909 (cliques in the n X n black bishop graph).
%Y Cf. A295910 (cliques in the n X n white bishop graph).
%K nonn
%O 0,2
%A _Abdullahi Umar_, Dec 28 2010
%E Renamed the sequence using more standard and widely-used terminology, _James Mitchell_, May 19 2012
|
