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A111384
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a(n) = binomial(n,3) - binomial(floor(n/2),3) - binomial(ceiling(n/2),3).
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10
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0, 0, 0, 1, 4, 9, 18, 30, 48, 70, 100, 135, 180, 231, 294, 364, 448, 540, 648, 765, 900, 1045, 1210, 1386, 1584, 1794, 2028, 2275, 2548, 2835, 3150, 3480, 3840, 4216, 4624, 5049, 5508, 5985, 6498, 7030, 7600, 8190, 8820, 9471, 10164, 10879, 11638, 12420, 13248
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OFFSET
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0,5
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COMMENTS
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a(n) is the maximum number of open triangles in a simple, undirected graph with n vertices. - Eugene Lykhovyd, Oct 20 2018
a(n) is the maximum number of elements of the set T := {3} u (IN \ 3IN) that can be written as a sum of three distinct elements of an n-element subset of T, see arXiv link 2309.14840. - Markus Sigg, Sep 27 2023
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: x^3*(1+2*x)/ ((1+x)^2 * (x-1)^4). (End)
a(n) = (n-2)*n^2/8 for even n, a(n) = (n-2)*(n^2-1)/8 for odd n. - Markus Sigg, Sep 26 2023
Sum_{n>=3} 1/a(n) = 4/3 - Pi^2/6 + 8*log(2)/3. - Amiram Eldar, Oct 10 2023
E.g.f.: (x + 2)*(x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x))/8. - Stefano Spezia, Apr 08 2024
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MAPLE
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MATHEMATICA
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LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 0, 0, 1, 4, 9}, 50] (* Vincenzo Librandi, Oct 20 2018 *)
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PROG
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(Magma) [Binomial(n, 3) - Binomial(Floor(n/2), 3) - Binomial(Ceiling(n/2), 3): n in [0..50]]; // Vincenzo Librandi, Oct 20 2018
(GAP) a:=[0, 0, 0, 1, 4, 9];; for n in [7..50] do a[n]:=2*a[n-1]+a[n-2]-4*a[n-3]+a[n-4]+2*a[n-5]-a[n-6]; od; a; # Muniru A Asiru, Oct 21 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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