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A173691
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Partial sums of round(n^2/6).
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2
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0, 0, 1, 3, 6, 10, 16, 24, 35, 49, 66, 86, 110, 138, 171, 209, 252, 300, 354, 414, 481, 555, 636, 724, 820, 924, 1037, 1159, 1290, 1430, 1580, 1740, 1911, 2093, 2286, 2490, 2706, 2934, 3175, 3429, 3696, 3976, 4270, 4578, 4901, 5239, 5592, 5960, 6344, 6744, 7161
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} round(k^2/6).
a(n) = round((2*n^3 + 3*n^2 + 6*n)/36).
a(n) = round((4*n^3 + 6*n^2 + 12*n + 5)/72).
a(n) = floor((2*n^3 + 3*n^2 + 6*n + 16)/36).
a(n) = ceiling((2*n^3 + 3*n^2 + 6*n - 11)/36).
a(n) = a(n-6) + n^2 - 5*n + 10, n > 5.
G.f.: x^2*(1+x^4)/((1+x)*(1-x+x^2)*(1+x+x^2)*(1-x)^4). - Bruno Berselli, Jan 12 2011
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EXAMPLE
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a(5) = round(1/6) + round(4/6) + round(9/6) + round(16/6) + round(25/6) = 0 + 1 + 2 + 3 + 4.
Note that 9/6 = 1.5 is rounded up.
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MAPLE
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a(n):=round((2*n^(3)+3*n^(2)+6*n)/(36))
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MATHEMATICA
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Accumulate[Round[Range[0, 50]^2/6]] (* or *) LinearRecurrence[{3, -3, 1, 0, 0, 1, -3, 3, -1}, {0, 0, 1, 3, 6, 10, 16, 24, 35}, 60] (* Harvey P. Dale, Jan 08 2014 *)
CoefficientList[Series[x^2(1+x^4)/((1+x)(1-x+x^2)(1+x+x^2)(1-x)^4), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 26 2014 *)
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PROG
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(Magma) [Floor((2*n^3+3*n^2+6*n+16)/36): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011
(PARI) vector(60, n, n--; (16+6*n+3*n^2+2*n^3)\36) \\ G. C. Greubel, Jul 02 2019
(Sage) [floor((16+6*n+3*n^2+2*n^3)/36) for n in (0..60)] # G. C. Greubel, Jul 02 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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