LinearRecurrence[{1, 2}, {24, 72}, 30] (* Harvey P. Dale, Jan 25 2020 *)

approved

editing

24, 24, 72, 120, 264, 504, 1032, 2040, 4104, 8184, 16392, 32760, 65544, 131064, 262152, 524280, 1048584, 2097144, 4194312, 8388600, 16777224, 33554424, 67108872, 134217720, 268435464, 536870904, 1073741832, 2147483640, 4294967304

3,4,1

Equivalently, this sequence also represents the sequence of sizes of the Graph of Vertex Colorings ('Coloring Graph') for 4-colorings of an n-wheel graph (related to another sequence on 3-colorings cf. A309314)

(Python)

import networkx as nx

from tqdm import tqdm

from libcolgraph import *

def wheelgraph(n):

'''

this kind of graph has $n$ vertices, one of them a 'central' vertex. $n-1$ vertices form a ring,

and the central vertex connects to each of the $n$ vertices to complete the spokes of the wheel.

'''

g = BaseGraph()

g.load_from_nx(nx.wheel_graph(n))

return g

def make_sequence(graphgen, *args, k=4, low=3, high=15, **kwargs):

'''

a function that accepts a graph generating function to generate the appropriate basegraph for parameter

n from low to high, and then calls 'build_coloring_graph' on it with parameter k, the number of colors

'''

for n in tqdm(range(low, high)):

g = graphgen(n, *args, **kwargs)

c = g.build_coloring_graph(k)

yield len(c)

[*make_sequence(wheelgraph, k=3)]

nonn,changed

nonn

Reduced offset from 4 to 3 from Aalok Sathe, Jul 23 2019

Added an additional term by Aalok Sathe, Jul 23 2019

editing

approved

proposed

editing

editing

proposed

Equivalently, this sequence also represents the sequence of sizes of the Graph of Vertex Colorings ('Coloring Graph') for 4-colorings of an n-wheel graph (related to another sequence on 3-colorings cf. A309314)

(Python)

import networkx as nx

from tqdm import tqdm

from libcolgraph import *

def wheelgraph(n):

'''

this kind of graph has $n$ vertices, one of them a 'central' vertex. $n-1$ vertices form a ring,

and the central vertex connects to each of the $n$ vertices to complete the spokes of the wheel.

'''

g = BaseGraph()

g.load_from_nx(nx.wheel_graph(n))

return g

def make_sequence(graphgen, *args, k=4, low=3, high=15, **kwargs):

'''

a function that accepts a graph generating function to generate the appropriate basegraph for parameter

n from low to high, and then calls 'build_coloring_graph' on it with parameter k, the number of colors

'''

for n in tqdm(range(low, high)):

g = graphgen(n, *args, **kwargs)

c = g.build_coloring_graph(k)

yield len(c)

[*make_sequence(wheelgraph, k=3)]

24, 24, 72, 120, 264, 504, 1032, 2040, 4104, 8184, 16392, 32760, 65544, 131064, 262152, 524280, 1048584, 2097144, 4194312, 8388600, 16777224, 33554424, 67108872, 134217720, 268435464, 536870904, 1073741832, 2147483640, 4294967304

4,3,1

Reduced offset from 4 to 3 from Aalok Sathe, Jul 23 2019

Added an additional term by Aalok Sathe, Jul 23 2019

approved

editing

proposed

approved

editing

proposed

The number of ways of m-coloring an annulus consisting of n zones joined like a pearl necklace is (m-1)^n + (-1)^n*(m-1), where m >= 3 (cf. A092297 for m=3). Now we must also color the central region.

O.g.f.: -24*x^3 - 12*x + 6 - 8/(1+x) - 2/(2*x-1). - R. J. Mathar, Dec 02 2007

a(n) = 24*A001045(n-2). [From _- _R. J. Mathar_, Aug 30 2008]

a(n) = 2^(n+1) - 8*(-1)^n. - _Vincenzo Librandi, _, Oct 10 2011

(MAGMA) [2^(n+1)-8*(-1)^n: n in [4..35]]; // _Vincenzo Librandi, _, Oct 10 2011

approved

editing

<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (1,2).