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A056827
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a(n) = floor(n^2/6).
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19
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0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 28, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 96, 104, 112, 121, 130, 140, 150, 160, 170, 181, 192, 204, 216, 228, 240, 253, 266, 280, 294, 308, 322, 337, 352, 368, 384, 400, 416, 433, 450, 468, 486, 504
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OFFSET
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0,5
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COMMENTS
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a(n-1) represents the floor of the area under the polygon connecting the lattice points (n, floor(n/3)) from 0..n, n>0 (see example). - Wesley Ivan Hurt, Jun 06 2014
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LINKS
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FORMULA
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G.f.: x^3*(1+x^2)/((1+x)*(1-x)^3*(1+x+x^2)*(1-x+x^2)).
a(n) = floor( (1/2) * Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1) ). - Wesley Ivan Hurt, Jun 06 2014
Sum_{n>=3} 1/a(n) = 15/8 + Pi^2/36 - Pi/(4*sqrt(3)) + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Aug 13 2022
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EXAMPLE
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0|.__.__./_|__|__|__|__|__|__|__|__|__|__|__|__|__|__|_________________
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 .. n
0 0 0 1 2 4 6 8 10 13 16 20 24 28 32 37 42 48 .. a(n)
0 0 0 1 2 4 6 8 10 13 16 20 24 28 32 37 42 .. a(n-1) <--
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MAPLE
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MATHEMATICA
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Floor[Range[0, 60]^2/6] (* or *) LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 1}, {0, 0, 0, 1, 2, 4, 6, 8}, 60] (* Harvey P. Dale, Jun 06 2013 *)
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PROG
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(Sage) [floor(n^2/6) for n in (0..60)] # G. C. Greubel, Jul 23 2019
(GAP) List([0..60], n-> Int(n^2/6) ); # G. C. Greubel, Jul 23 2019
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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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