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A144679
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a(n) = [n/5 + 1]*[n/5 + 2]*(3*n - 10*[n/5] + 3)/6, where [.] = floor.
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5
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1, 2, 3, 4, 5, 8, 11, 14, 17, 20, 26, 32, 38, 44, 50, 60, 70, 80, 90, 100, 115, 130, 145, 160, 175, 196, 217, 238, 259, 280, 308, 336, 364, 392, 420, 456, 492, 528, 564, 600, 645, 690, 735, 780, 825, 880, 935, 990, 1045, 1100, 1166, 1232, 1298, 1364, 1430, 1508, 1586, 1664
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OFFSET
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0,2
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COMMENTS
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Related to enumeration of quantum states: this is S_c defined in eq.(10b) of the O'Sullivan and Busch reference, with lambda = 5.
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
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,2,-4,2,0,0,-1,2,-1).
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FORMULA
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a(n-4) + a(n-3) + a(n-2) + a(n-1) + a(n) = A122047(n+2).
G.f.: 1/((1-x)^4*(1 + x + x^2 + x^3 + x^4)^2). (End)
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
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MAPLE
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n:=80; lambda:=5; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b), snc];
od:
S10b;
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
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MATHEMATICA
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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 *)
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 *)
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PROG
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(PARI) apply( {A144679(n)=(3*n+3-10*n\=5)*(n+1)*(n+2)\6}, [0..55]) \\ M. F. Hasler, Jun 25 2021
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^5))^2 )); // G. C. Greubel, Oct 18 2021
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/((1-x)*(1-x^5))^2 ).list()
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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