OFFSET
0,2
COMMENTS
Also the number of (not necessarily maximal) 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) = 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
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