A152390 Arises in enumerating non-degenerate colorings in Brook's Theorem.

6, 68, 252, 648, 1370, 2556, 4368, 6992, 10638, 15540, 21956, 30168, 40482, 53228, 68760, 87456, 109718, 135972, 166668, 202280, 243306, 290268, 343712, 404208, 472350, 548756, 634068, 728952, 834098, 950220, 1078056, 1218368, 1371942

Offset

0,1

Comments

Gravin's abstract: Let c => 2 and p => c be two integers. We will call a proper coloring of the graph G a (c,p)-nondegenerate, if for any vertex of G with degree at least p there are at least c vertices of different colors adjacent to it. In our work we prove the following result, which generalizes Brook's Theorem. Let D => 3 and G be a graph without cliques on D+1 vertices and the degree of any vertex in this graph is not greater than D. Then for every integer c => 2 there is a proper (c,p)-nondegenerate vertex D-coloring of G, where p=(c^3+8c^2+19c+6)(c+1). During the primary proof, some interesting corollaries are derived.

Links

Table of n, a(n) for n=0..32.

Nikolay Gravin, Non-degenerate colorings in the Brook's Theorem, Dec 1, 2008.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (n^3+8*n^2+19*n+6)*(n+1).

G.f.: 2(3+19x-14x^2+4x^3)/(1-x)^5. [From R. J. Mathar, Dec 04 2008]

Example

a(2) = 252 because ((2^3) + (8 * (2^2)) + (19 * 2) + 6) * (2 + 1) = 252.

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {6, 68, 252, 648, 1370}, 50] (* Harvey P. Dale, Jan 29 2023 *)

Crossrefs

Sequence in context: A116005 A297435 A281058 * A200059 A183470 A281051

Adjacent sequences: A152387 A152388 A152389 * A152391 A152392 A152393

Keyword

easy,nonn

Author

Jonathan Vos Post, Dec 03 2008

Status

approved