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A183156 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). 3
1, 2, 7, 22, 59, 142, 319, 686, 1435, 2950, 5999, 12118, 24379, 48926, 98047, 196318, 392891, 786070, 1572463, 3145286, 6290971, 12582382, 25165247, 50331022, 100662619, 201325862, 402652399, 805305526, 1610611835, 3221224510, 6442449919 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also the number of (not necessarily maximum) cliques in the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.

Eric Weisstein's World of Mathematics, Bishop Graph

Eric Weisstein's World of Mathematics, Clique

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

FORMULA

T(n) = 3*2^(n+1) - (n+2)^2 - 1, (n >= 0).

G.f.: (1 - 3*x + 6*x^2 - 2*x^3) / ( (2*x - 1)*(x - 1)^3 ). - R. J. Mathar, Jul 03 2011

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4). - Eric W. Weisstein, Nov 29 2017

a(n) = A295909(n) + A295910(n) for n > 1. - Eric W. Weisstein, Nov 29 2017

a(n) = 2*a(n-1) + n^2 - 1. - Kritsada Moomuang, Oct 25 2019

EXAMPLE

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.

MATHEMATICA

LinearRecurrence[{5, -9, 7, -2}, {1, 2, 7, 22}, 40] (* Vincenzo Librandi, Oct 11 2017 *)

Table[3 2^(n + 1) - (n + 2)^2 - 1, {n, 0, 30}] (* Vincenzo Librandi, Oct 11 2017 *)

LinearRecurrence[{5, -9, 7, -2}, {2, 7, 22, 59}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)

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 *)

PROG

(PARI) for(n=0, 33, print1(3*(2^(n+1))-(n+2)^2-1, ", "))

(MAGMA) [3*2^(n+1)-(n+2)^2-1: n in [0..33]]; // Vincenzo Librandi, Oct 11 2017

CROSSREFS

Row sums of triangles A183157, A183158.

Cf. A295909 (cliques in the n X n black bishop graph).

Cf. A295910 (cliques in the n X n white bishop graph).

Sequence in context: A212384 A306347 A063019 * A018039 A198888 A084264

Adjacent sequences:  A183153 A183154 A183155 * A183157 A183158 A183159

KEYWORD

nonn

AUTHOR

Abdullahi Umar, Dec 28 2010

EXTENSIONS

Renamed the sequence using more standard and widely-used terminology, James Mitchell, May 19 2012

STATUS

approved

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Last modified January 26 11:07 EST 2020. Contains 331279 sequences. (Running on oeis4.)