A140986 -id:A140986 - cp's OEIS frontend

A115400 Number of n-colorings of the octahedral graph.

0, 0, 0, 6, 96, 780, 4080, 15330, 45696, 115416, 257760, 523710, 987360, 1752036, 2957136, 4785690, 7472640, 11313840, 16675776, 24006006, 33844320, 46834620, 63737520, 85443666, 112987776, 147563400, 190538400, 243471150, 308127456

Offset: 0

Jonathan Vos Post, Aug 25 2008

The octahedral graph is the dual of the cubical graph whose chromatic polynomial is evaluated in A140986.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Octahedral Graph.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A140986.

[n*(n-1)*(n-2)*(n^3 - 9*n^2 + 29*n - 32): n in [0..50]]; // Vincenzo Librandi, Feb 12 2012

Table[n*(n-1)*(n-2)*(n^3-9*n^2+29*n-32),{n,0,50}] (* Vincenzo Librandi, Feb 12 2012 *)

A115400(n):=n*(n-1)*(n-2)*(n^3 - 9*n^2 + 29*n - 32)$
makelist(A115400(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n) = n*(n-1)*(n-2)*(n^3 - 9*n^2 + 29*n - 32).
G.f.: 6*x^3*(1 + 9*x + 39*x^2 + 71*x^3)/(1-x)^7. - Colin Barker, Feb 12 2012
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 6. - Chai Wah Wu, Jan 19 2024

A334159 Irregular triangle read by rows: T(n,k) is the number of colorings of the n-hypercube graph using exactly k unlabeled colors, k = 1..2^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 1, 18, 92, 146, 80, 16, 1, 0, 1, 494, 54583, 1507094, 12630906, 40096740, 58031885, 43419502, 18212138, 4498756, 670366, 60220, 3156, 88, 1, 0, 1, 197546, 5427041958, 17973998149410, 10961517110194516, 1450479305675145412, 56507865332978414188

Offset: 0

Andrew Howroyd, Apr 21 2020

			Triangle begins:
0 | 1;
1 | 0, 1;
2 | 0, 1, 2, 1;
3 | 0, 1, 18, 92, 146, 80, 16, 1;
4 | 0, 1, 494, 54583, 1507094, 12630906, 40096740, 58031885, 43419502, 18212138, 4498756, 670366, 60220, 3156, 88, 1;

Andrew Howroyd, Table of n, a(n) for n = 0..62 (rows 0..5)
Andrew Howroyd, Chromatic Polynomials of Hypercubes Q_0 to Q_5
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Eric Weisstein's World of Mathematics, Hypercube Graph

Cf. A091940, A140986, A158348, A296914, A295176, A334247.

A158348 Number of n-colorings of the Hypercube Graph Q4.

Original entry on oeis.org

0, 0, 2, 2970, 1321860, 187430900, 10199069190, 269591166222, 4221404762120, 44876701584360, 355148098691850, 2230178955481730, 11630998385335692, 52097117078470620, 205557074788375310, 728566149746575350, 2355657801908655120, 7034253747275048912, 19594719516430397970

Offset: 0

Alois P. Heinz, Mar 16 2009

The Hypercube Graph Q4 has 16 vertices and 32 edges.

All terms are even.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Timme, Marc; van Bussel, Frank; Fliegner, Denny; Stolzenberg, Sebastian (2009) "Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions", New J. Phys. 11 023001, doi: 10.1088/1367-2630/11/2/023001.
Eric Weisstein's World of Mathematics, Hypercube Graph
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A140986, A380589.

Column k=4 of A342128.

a:= n-> n^16 -32*n^15 +496*n^14 -4936*n^13 +35264*n^12 -191600*n^11 +818036*n^10 -2794896*n^9 +7701952*n^8 -17100952*n^7 +30276984*n^6 -41821924*n^5 +43389646*n^4 -31680240*n^3 +14412776*n^2 -3040575*n:
seq(a(n), n=0..20);

a(n) = n^16 -32*n^15 + ... (see Maple program).

A268283 Number of distinct directed Hamiltonian cycles of the Platonic graphs (in the order of tetrahedral, cubical, octahedral, dodecahedral, and icosahedral graph).

Original entry on oeis.org

6, 12, 32, 60, 2560

Offset: 1

Melvin Peralta, Jan 29 2016

a(n)/2 is the number of distinct undirected Hamiltonian cycles of the Platonic graph corresponding to a(n).

Eric Weisstein's World of Mathematics, Tetrahedral Graph
Eric Weisstein's World of Mathematics, Cubical Graph
Eric Weisstein's World of Mathematics, Octahedral Graph
Eric Weisstein's World of Mathematics, Dodecahedral Graph
Eric Weisstein's World of Mathematics, Icosahedral Graph

Cf. A052762 (tetrahedral graph), A140986 (cubical graph), A115400 (octahedral graph), A218513 (dodecahedral graph), A218514 (icosahedral graph).

A334247 Number of acyclic orientations of the edges of an n-dimensional cube.

Original entry on oeis.org

1, 2, 14, 1862, 193270310, 47171704165698393638

Offset: 0

Matthew Scroggs, Apr 20 2020

a(n) is the absolute value of the chromatic polynomial of the n-hypercube graph evaluated at -1.

			For n=2, there are 14 ways to orient the edges of a square without cycles (see links).

David Eppstein, 14 acyclic orientations of a square
Eric Weisstein's World of Mathematics, Hypercube Graph

Cf. A334248 is the number of acyclic orientations with rotations and reflections of the same orientation excluded.
Cf. A140986, A158348, A296914, A334159, A334278.
Cf. A033815 (cross-polytope), A058809 (wheel), A338152 (demihypercube), A338153 (prism), A338154 (antiprism).

with(GraphTheory): with(SpecialGraphs):
a:= n-> abs(ChromaticPolynomial(HypercubeGraph(n), -1)):
seq(a(n), n=0..4);  # Alois P. Heinz, Jan 14 2025

a(n) = Sum_{k=1..2^n} (-1)^(2^n-k) * k! * A334159(n, k). - Andrew Howroyd, Apr 21 2020

a(n) = |Sum_{k=0..2^n} (-1)^k * A334278(n, k)|. - Peter Kagey, Oct 13 2020

a(5) from Andrew Howroyd, Apr 23 2020

A218514 Number of n-colorings of the icosahedral graph.

Original entry on oeis.org

0, 0, 0, 0, 240, 80400, 4012560, 76848240, 825447840, 6005512800, 33014872800, 146953113120, 554770648080, 1835249610480, 5448481998960, 14778817981200, 37135461679680, 87386816771520, 194264943433920, 410876964198720, 831638579799600, 1618744884780240

Offset: 0

Eric M. Schmidt, Oct 31 2012

N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. See p. 69.

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
Eric W. Weisstein, Icosahedral Graph
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A052762, A140986, A115400, A218513, A296916, A296917..

A218514(n):=n*(n-1)*(n-2)*(n-3)*(n^8 -24*n^7 +260*n^6 -1670*n^5 +6999*n^4 -19698*n^3 +36408*n^2 -40240*n +20170)$
makelist(A218514(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

def A218514(n) : return n*(n-1)*(n-2)*(n-3)*(n^8 -24*n^7 +260*n^6 -1670*n^5 +6999*n^4 -19698*n^3 +36408*n^2 -40240*n +20170);

a(n) = n(n-1)(n-2)(n-3)(n^8 -24n^7 +260n^6 -1670n^5 +6999n^4 -19698n^3 +36408n^2 -40240n +20170).
Hence a(n) = n^12 - 30*n^11 + 415*n^10 - 3500*n^9 + 20023*n^8 - 81622*n^7 + 241605*n^6 - 517360*n^5 + 780286*n^4 - 782108*n^3 + 463310*n^2 - 121020*n (cf. A296917) - N. J. A. Sloane, Dec 23 2017
G.f.: -240*x^4*(12547*x^8 +131518*x^7 +481078*x^6 +743494*x^5 +485740*x^4 +128698*x^3 +12442*x^2 +322*x +1)/(x-1)^13. [Colin Barker, Nov 06 2012]

A334278 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the cubical graph Q_n, 0 <= k <= 2^n.

Original entry on oeis.org

0, 1, 0, -1, 1, 0, -3, 6, -4, 1, 0, -133, 423, -572, 441, -214, 66, -12, 1, 0, -3040575, 14412776, -31680240, 43389646, -41821924, 30276984, -17100952, 7701952, -2794896, 818036, -191600, 35264, -4936, 496, -32, 1

Offset: 0

Peter Kagey, Apr 21 2020

The sums of the absolute values of the entries in each row gives A334247, the number of acyclic orientations of edges of the n-cube.

			Table begins:
n/k| 0     1    2     3    4     5   6    7  8
---+-------------------------------------------
  0| 0,    1
  1| 0,   -1,   1
  2| 0,   -3,   6,   -4,   1
  3| 0, -133, 423, -572, 441, -214, 66, -12, 1

Peter Kagey, Table of n, a(n) for n = 0..68 (rows 0..5; row 5 from Andrew Howroyd's file)
Andrew Howroyd, Chromatic Polynomials of Hypercubes Q_0 to Q_5
Eric Weisstein's World of Mathematics, Chromatic Polynomial.
Eric Weisstein's World of Mathematics, Hypercube Graph.

Cf. A296914 is the reverse of row 3.
Cf. A334279 is analogous for the n-dimensional cross-polytope, the dual of the n-cube.
Cf. A001787, A002378, A091940, A140986, A158348.
Cf. A334159, A334247, A334358.

with(GraphTheory): with(SpecialGraphs):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
    ChromaticPolynomial(HypercubeGraph(n), x)):
seq(T(n), n=0..4);  # Alois P. Heinz, Jan 14 2025

T[n_, k_] := Coefficient[ChromaticPolynomial[HypercubeGraph[n], x], x, k]

T(n,0) = 0.

T(n,k) = Sum_{i=1..2^n}, Stirling1(i,k) * A334159(n,i). - Andrew Howroyd, Apr 25 2020

A218513 Number of n-colorings of the dodecahedral graph.

Original entry on oeis.org

0, 0, 0, 7200, 168506880, 112603286160, 15108392957760, 775405390866960, 20886647215714560, 353998543659193440, 4231366997071432320, 38508081275604409920, 281586666065022616320, 1720887594454493527920, 9053942417801770507200, 41955877772610102690480

Offset: 0

Eric M. Schmidt, Oct 31 2012

N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. See pp. 69-70.

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
Eric W. Weisstein, Dodecahedral Graph
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Cf. A052762, A140986, A115400, A218514, A296918, A296919.

A218513(n):=n*(n-1)*(n-2)*(n^17 -27*n^16 +352*n^15 -2950*n^14 +17839*n^13 -82777*n^12 +305866*n^11 -921448*n^10 +2297495*n^9 -4783425*n^8 +8347700*n^7 -12195590*n^6 +14808795*n^5 -14713381*n^4 +11613602*n^3 -6892084*n^2 +2751604*n -555984)$
makelist(A218513(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

def A218513(n) : return n*(n-1)*(n-2)*(n^17 -27*n^16 +352*n^15 -2950*n^14 +17839*n^13 -82777*n^12 +305866*n^11 -921448*n^10 +2297495*n^9 -4783425*n^8 +8347700*n^7 -12195590*n^6 +14808795*n^5 -14713381*n^4 +11613602*n^3 -6892084*n^2 +2751604*n -555984);

a(n) = n(n-1)(n-2)(n^17 - 27n^16 + 352n^15 - 2950n^14 + 17839n^13 - 82777n^12 + 305866n^11 - 921448n^10 + 2297495n^9 - 4783425n^8 + 8347700n^7 - 12195590n^6 + 14808795n^5 - 14713381n^4 + 11613602n^3 - 6892084n^2 + 2751604n - 555984).
See A296919 for the coefficients of the expanded form of a(n). - N. J. A. Sloane, Dec 23 2017
G.f.: -240*x^3*(2007273*x^17 +678113783*x^16 +62897280675*x^15 +2149103163405*x^14 +32571452423195*x^13 +246267894384141*x^12 +1000326687571911*x^11 +2283861589692665*x^10 +3002531231655465*x^9 +2288662487004975*x^8 +1001857651156729*x^7 +244960098705399*x^6 +31779760521705*x^5 +2006465657455*x^4 +53246253405*x^3 +454442307*x^2 +701482*x +30)/(x-1)^21. - Colin Barker, Nov 06 2012

A296914 List of coefficients of chromatic polynomial of the cubical graph Q_3, highest order terms first.

Original entry on oeis.org

1, -12, 66, -214, 441, -572, 423, -133, 0

Offset: 1

N. J. A. Sloane, Dec 22 2017

Eric Weisstein's World of Mathematics, Chromatic Polynomial
Eric Weisstein's World of Mathematics, Cubical Graph

Cf. A140986, A334159.

The chromatic polynomial is x^8-12*x^7+66*x^6-214*x^5+441*x^4-572*x^3+423*x^2-133*x.

a(n) = Sum_{k=1..8} Stirling1(k, 9-n)*A334159(3,k). - Andrew Howroyd, Apr 22 2020

A334356 Number of nonequivalent proper colorings of the vertices of a cube using at most n colors up to rotations and reflections of the cube.

Original entry on oeis.org

0, 1, 15, 154, 1115, 5955, 24836, 85260, 251154, 655005, 1548085, 3374646, 6876805, 13237679, 24271170, 42667640, 72305556, 118640025, 189179979, 294066610, 446766495, 664893691, 971175920, 1394580804, 1971618950, 2747841525, 3779550801, 5135742990, 6900303529

Offset: 1

Andrew Howroyd, Apr 24 2020

Adjacent vertices may not have the same color.

a(n) is the number of nonequivalent n-colorings of the cubical graph up to graph isomorphism.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cubical Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A128766, A140986, A334357, A334358.

a(n) = {n*(n - 1)*(n^6 - 11*n^5 + 61*n^4 - 195*n^3 + 384*n^2 - 428*n + 216)/48}

a(n) = n*(n - 1)*(n^6 - 11*n^5 + 61*n^4 - 195*n^3 + 384*n^2 - 428*n + 216)/48.

a(n) = Sum_{k=1..8} n^k * A334358(3,8-k) / 48.

cp's OEIS Frontend

A115400 Number of n-colorings of the octahedral graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

Formula

A334159 Irregular triangle read by rows: T(n,k) is the number of colorings of the n-hypercube graph using exactly k unlabeled colors, k = 1..2^n.

Views

Author

Keywords

Examples

Links

Crossrefs

A158348 Number of n-colorings of the Hypercube Graph Q4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A268283 Number of distinct directed Hamiltonian cycles of the Platonic graphs (in the order of tetrahedral, cubical, octahedral, dodecahedral, and icosahedral graph).

Views

Author

Keywords

Comments

Links

Crossrefs

A334247 Number of acyclic orientations of the edges of an n-dimensional cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A218514 Number of n-colorings of the icosahedral graph.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maxima

Sage

Formula

A334278 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the cubical graph Q_n, 0 <= k <= 2^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A218513 Number of n-colorings of the dodecahedral graph.

Views

Author

Keywords

References

Links

Crossrefs

Programs