A006528 a(n) = (n^4 + n^2 + 2*n)/4. (Formerly M4160)

0, 1, 6, 24, 70, 165, 336, 616, 1044, 1665, 2530, 3696, 5226, 7189, 9660, 12720, 16456, 20961, 26334, 32680, 40110, 48741, 58696, 70104, 83100, 97825, 114426, 133056, 153874, 177045, 202740, 231136, 262416, 296769, 334390, 375480, 420246, 468901, 521664, 578760

Offset

0,3

Comments

Number of ways to color vertices of a square using <= n colors, allowing only rotations.

Also product of first and last terms in n-th row of a triangle of form: row(1)= 1; row(2)= 2,3; row(3) = 4, 5, 6, ... . - Dave Durgin, Aug 17 2012

References

Nick Baxter, The Burnside di-lemma: combinatorics and puzzle symmetry, in Tribute to a Mathemagician, Peters, 2005, pp. 199-210.

M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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 (5,-10,10,-5,1).

Formula

a(n) = n*(n+1)*(n^2-n+2)/4.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Apr 30 2012

From Stefano Spezia, Oct 11 2023: (Start)

O.g.f.: x*(1 + x + 4*x^2)/(1 - x)^5.

E.g.f.: exp(x)*x*(4 + 8*x + 6*x^2 + x^3)/4. (End)

Maple

A006528:=-z*(1+z+4*z**2)/(z-1)**5; # Simon Plouffe in his 1992 dissertation
a:=n->add(n+add(binomial(n, 2), j=1..n), j=0..n):seq(a(n)/2, n=0..35); # Zerinvary Lajos, Aug 26 2008

Mathematica

Table[CycleIndex[CyclicGroup[4], t]/.Table[t[i]->n, {i, 1, 4}], {n, 0, 20}] (* Geoffrey Critzer, Mar 13 2011*)
Table[(n^4+n^2+2*n)/4, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 6, 24, 70}, 40] (* Harvey P. Dale, Jan 13 2019 *)

Prog

(Magma) I:=[0, 1, 6, 24, 70]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2012
(PARI) a(n) = n*(n+1)*(n^2-n+2)/4; /* Joerg Arndt, Apr 30 2012 */

Crossrefs

Row n=2 of A343095.

Cf. A002817 (square colorings).

Sequence in context: A274772 A234271 A338512 * A052749 A337021 A262445

Adjacent sequences: A006525 A006526 A006527 * A006529 A006530 A006531

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved