A006565 Number of ways to color vertices of a hexagon using <= n colors, allowing only rotations. (Formerly M4942)

0, 1, 14, 130, 700, 2635, 7826, 19684, 43800, 88725, 166870, 295526, 498004, 804895, 1255450, 1899080, 2796976, 4023849, 5669790, 7842250, 10668140, 14296051, 18898594, 24674860, 31853000, 40692925, 51489126, 64573614, 80318980, 99141575, 121504810, 147922576, 178962784

Offset

0,3

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Victor Meally, Letter to N. J. A. Sloane, no date.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = n*(n+1)*(n^2+n+1)*(n^2-2*n+2)/6.

Maple

A006565 := n-> (n^6+n^3+2*n^2+2*n)/6;
# Alternative:
G006565 := -(1+7*z+53*z**2+49*z**3+10*z**4)/(z-1)**7; # Conjectured by Simon Plouffe in his 1992 dissertation.

Mathematica

A006565[n_] := n*(n*(n^4 + n + 2) + 2)/6; Array[A006565, 35, 0] (* Paolo Xausa, May 17 2026 *)

Crossrefs

Cf. A027670, A075195.

Sequence in context: A240189 A021354 A208427 * A206207 A358168 A026936

Adjacent sequences: A006562 A006563 A006564 * A006566 A006567 A006568

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Paolo Xausa, May 17 2026

Status

approved