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A006528
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a(n) = (n^4 + n^2 + 2*n)/4.
(Formerly M4160)
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16
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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).
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FORMULA
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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]
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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