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 A006528 a(n) = (n^4 + n^2 + 2*n)/4. (Formerly M4160) 10

%I M4160

%S 0,1,6,24,70,165,336,616,1044,1665,2530,3696,5226,7189,9660,12720,

%T 16456,20961,26334,32680,40110,48741,58696,70104,83100,97825,114426,

%U 133056,153874,177045,202740,231136,262416,296769,334390,375480

%N a(n) = (n^4 + n^2 + 2*n)/4.

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

%C 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

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

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

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

%H Vincenzo Librandi, <a href="/A006528/b006528.txt">Table of n, a(n) for n = 0..1000</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

%F 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]

%p A006528:=-z*(1+z+4*z**2)/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation

%p 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

%t Table[CycleIndex[CyclicGroup[4],t]/.Table[t[i]->n,{i,1,4}],{n,0,20}] (* _Geoffrey Critzer_, Mar 13 2011*)

%t 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 *)

%o (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

%o (PARI) a(n) = n*(n+1)*(n^2-n+2)/4; /* _Joerg Arndt_, Apr 30 2012 */

%Y Cf. A002817 (square colorings).

%K nonn,easy,changed

%O 0,3

%A _N. J. A. Sloane_

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Last modified January 18 21:54 EST 2019. Contains 319282 sequences. (Running on oeis4.)