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a(n) = [n/5 + 1]*[n/5 + 2]*(3*n - 10*[n/5] + 3)/6, where [.] = floor.
5

%I #51 Sep 08 2022 08:45:38

%S 1,2,3,4,5,8,11,14,17,20,26,32,38,44,50,60,70,80,90,100,115,130,145,

%T 160,175,196,217,238,259,280,308,336,364,392,420,456,492,528,564,600,

%U 645,690,735,780,825,880,935,990,1045,1100,1166,1232,1298,1364,1430,1508,1586,1664

%N a(n) = [n/5 + 1]*[n/5 + 2]*(3*n - 10*[n/5] + 3)/6, where [.] = floor.

%C Related to enumeration of quantum states: this is S_c defined in eq.(10b) of the O'Sullivan and Busch reference, with lambda = 5.

%C This coincides with the formula for an upper bound on the minimum number of monochromatic triangles in the complete graph K_{n+11} with 3-colored edges given by Cummings et al. (2013) in Corollary 3. (The paper claims that this bound is sharp only for all sufficiently large n.) - _M. F. Hasler_, Jun 25 2021

%H G. C. Greubel, <a href="/A144679/b144679.txt">Table of n, a(n) for n = 0..1000</a>

%H James Cummings, Daniel Král', Florian Pfender, Konrad Sperfeld, Andrew Treglown, and Michael Young, <a href="https://doi.org/10.1016/j.jctb.2013.05.002">Monochromatic triangles in three-coloured graphs</a>, Journal of Combinatorial Theory B 103 no. 4 (2013) 489-503 (also: arXiv:1206.1987).

%H Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231v1 [quant-ph], 2008. [Eq (10b), lambda=5]

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,2,-4,2,0,0,-1,2,-1).

%F From _Johannes W. Meijer_, May 20 2011: (Start)

%F a(n-4) + a(n-3) + a(n-2) + a(n-1) + a(n) = A122047(n+2).

%F G.f.: 1/((1-x)^4*(1 + x + x^2 + x^3 + x^4)^2). (End)

%F a(n) = r*A000292(q+1) + (5-r)*A000292(q) = (n + 2r + 1)*(q + 2)*(q + 1)/6, where A000292(q) = binomial(q+2,3), r = (n+1) mod 5, q = (n+1-r)/5. - _M. F. Hasler_, Jun 25 2021

%p n:=80; lambda:=5; S10b:=[];

%p for ii from 0 to n do

%p x:=floor(ii/lambda);

%p snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);

%p S10b:=[op(S10b),snc];

%p od:

%p S10b;

%p A144679 := proc(n) option remember; local k; sum(THN5(n-k),k=0..4) end: THN5:= proc(n) option remember; THN5(n):= binomial(floor(n/5)+3,3) end: seq(A144679(n), n=0..57); # _Johannes W. Meijer_, May 20 2011

%t LinearRecurrence[{2,-1,0,0,2,-4,2,0,0,-1,2,-1}, {1,2,3,4,5,8,11,14,17,20,26,32}, 60] (* _Jean-François Alcover_, Nov 22 2017 *)

%t CoefficientList[Series[1/((x-1)^4(x^4+x^3+x^2+x+1)^2),{x,0,100}],x] (* _Harvey P. Dale_, Aug 29 2021 *)

%o (PARI) apply( {A144679(n)=(3*n+3-10*n\=5)*(n+1)*(n+2)\6}, [0..55]) \\ _M. F. Hasler_, Jun 25 2021

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^5))^2 )); // _G. C. Greubel_, Oct 18 2021

%o (Sage)

%o def A144679_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 1/((1-x)*(1-x^5))^2 ).list()

%o A144679_list(60) # _G. C. Greubel_, Oct 18 2021

%Y Cf. A000292, A006918, A122047, A144678, A144677.

%K easy,nonn

%O 0,2

%A _N. J. A. Sloane_, Feb 06 2009