A122695 Number of edges in the n-th Mycielski graph.

0, 0, 1, 5, 20, 71, 236, 755, 2360, 7271, 22196, 67355, 203600, 613871, 1847756, 5555555, 16691240, 50122871, 150466916, 451597355, 1355185280, 4066342271, 12200599676, 36604944755, 109821125720, 329475960071, 988453046036, 2965409469755, 8896329072560

Offset

0,4

Comments

The Mycielski graphs are formed, starting from the empty graph, by repeatedly applying a construction of Mycielski for generating triangle-free graphs with arbitrarily large chromatic number.

The number of vertices in the Mycielski graphs is given by sequence A083329.

Also the number of maximal and maximum cliques in the n-Mycielski graph for n > 1. - Eric W. Weisstein, Dec 01 2017

Links

Colin Barker, Table of n, a(n) for n = 0..1000

J. Mycielski, Sur le coloriage des graphes, Colloq. Math. 3: 161-162, 1955.

Eric Weisstein's World of Mathematics, Maximal Clique

Eric Weisstein's World of Mathematics, Maximum Clique

Eric Weisstein's World of Mathematics, Mycielski Graph

Emre Yolcu, Xinyu Wu, Marijn J. H. Heule, Mycielski graphs and PR proofs, Carnegie Mellon University (2020).

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Formula

a(n) = 3*a(n-1) + 3*2^(n-3) - 1 for n > 2. [corrected by Moeez Muhammad, Apr 17 2021]

From Colin Barker, Mar 07 2012 and Jan 16 2016: (Start)

a(n) = (18-27*2^n+14*3^n)/36 for n>1.

a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3) for n>4.

G.f.: x^2*(1-x+x^2) / ((1-x)*(1-2*x)*(1-3*x)).

(End)

Example

The first few graphs in the sequence of Mycielski graphs are the null graph, K1, K2, C5 and the Graezsch graph with 11 vertices and 20 edges. Thus the first entries in this sequence are 0, 0, 1, 5 and 20.

Mathematica

Table[If[n < 2, 0, (18 - 27 2^n + 14 3^n)/36], {n, 0, 10}] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0, 0}, LinearRecurrence[{6, -11, 6}, {1, 5, 20}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
ientList[Series[x^2 (1 - x + x^2)/(1 - 6 x + 11 x^2 - 6 x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

Prog

(PARI) concat(vector(2), Vec(x^2*(1-x+x^2)/((1-x)*(1-2*x)*(1-3*x)) + O(x^40))) \\ Colin Barker, Jan 16 2016

Crossrefs

Cf. A083329.

Sequence in context: A036683 A054444 A121332 * A269914 A066822 A137212

Adjacent sequences: A122692 A122693 A122694 * A122696 A122697 A122698

Keyword

easy,nonn

Author

David Eppstein, Oct 29 2006

Status

approved