cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Melvin Peralta, Jan 29 2016

Keywords

Comments

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

Links

Crossrefs

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

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

Views

Author

Eric M. Schmidt, Oct 31 2012

Keywords

References

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

Links

Crossrefs

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

Programs

  • Maxima
    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 */
  • Sage
    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);

Formula

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

A296916 List of coefficients of reduced chromatic polynomial of icosahedron, highest order terms first.

Original entry on oeis.org

1, -24, 260, -1670, 6999, -19698, 36408, -40240, 20170
Offset: 1

Views

Author

N. J. A. Sloane, Dec 22 2017

Keywords

Comments

These are the coefficients when the chromatic polynomial of the icosahedron (see A296917) is divided by x*(x-1)*(x-2)*(x-3).

Examples

			The reduced chromatic polynomial is x^8-24*x^7+260*x^6-1670*x^5+6999*x^4-19698*x^3+36408*x^2-40240*x+20170.
Multiplying by x*(x-1)*(x-2)*(x-3) and expanding we get the chromatic polynomial for the icosahedron, which is x^12 - 30*x^11 + 415*x^10 - 3500*x^9 + 20023*x^8 - 81622*x^7 + 241605*x^6 - 517360*x^5 + 780286*x^4 - 782108*x^3 + 463310*x^2 - 121020*x.

References

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

Crossrefs

Cf. A296917, A218514.

A296917 List of coefficients of chromatic polynomial of icosahedron, highest order terms first.

Original entry on oeis.org

1, -30, 415, -3500, 20023, -81622, 241605, -517360, 780286, -782108, 463310, -121020, 0
Offset: 1

Views

Author

N. J. A. Sloane, Dec 22 2017

Keywords

Examples

			The polynomial is x^12 - 30*x^11 + 415*x^10 - 3500*x^9 + 20023*x^8 - 81622*x^7 + 241605*x^6 - 517360*x^5 + 780286*x^4 - 782108*x^3 + 463310*x^2 - 121020*x.

References

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

Crossrefs

Cf. A296916, A218514.

A296919 List of coefficients of chromatic polynomial of dodecahedron, highest order terms first.

Original entry on oeis.org

1, -30, 435, -4060, 27393, -142194, 589875, -2004600, 5673571, -13518806, 27292965, -46805540, 68090965, -83530946, 85371335, -71159652, 46655060, -22594964, 7171160, -1111968, 0
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 2017

Keywords

Comments

This is based on A218513, not on the expression at the top of page 70 of the Biggs reference, which I could not get to work.

Examples

			The chromatic polynomial is x^20 - 30*x^19 + 435*x^18 - 4060*x^17 + 27393*x^16 - 142194*x^15 + 589875*x^14 - 2004600*x^13 + 5673571*x^12 - 13518806*x^11 + 27292965*x^10 - 46805540*x^9 + 68090965*x^8 - 83530946*x^7 + 85371335*x^6 - 71159652*x^5 + 46655060*x^4 - 22594964*x^3 + 7171160*x^2 - 1111968*x.

References

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

Crossrefs

Cf. A218513, A218514, A296918.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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