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A175842
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Partial sums of ceiling(n^2/14).
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1
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0, 1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 42, 53, 66, 80, 97, 116, 137, 161, 187, 216, 248, 283, 321, 363, 408, 457, 510, 566, 627, 692, 761, 835, 913, 996, 1084, 1177, 1275, 1379, 1488, 1603, 1724, 1850, 1983, 2122, 2267, 2419, 2577, 2742, 2914, 3093
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OFFSET
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0,3
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COMMENTS
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There are several sequences of integers of the form ceiling(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,8,10,11,12, 14,15,16,19,20,23,24).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).
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FORMULA
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a(n) = round((2*n+1)*(2*n^2 + 2*n + 45)/168).
a(n) = floor((2*n^3 + 3*n^2 + 46*n + 60)/84).
a(n) = ceiling((2*n^3 + 3*n^2 + 46*n - 15)/84).
a(n) = a(n-14) + (n+1)*(n-14) + 80, n > 13.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-14) - 3*a(n-15) + 3*a(n-16) - a(n-17). - R. J. Mathar, Mar 11 2012
G.f.: x*(x^14 - x^13 + x^11 - x^10 + x^9 + x^5 - x^4 + x^3 - x + 1)/((x-1)^4*(x+1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)). - Colin Barker, Oct 26 2012
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EXAMPLE
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a(14) = 0 + 1 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 + 14 = 80.
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MAPLE
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seq(ceil((2*n^3+3*n^2+46*n-15)/84), n=0..50)
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MATHEMATICA
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Accumulate[Ceiling[Range[0, 50]^2/14]] (* Harvey P. Dale, May 14 2017 *)
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PROG
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(Magma) [Round((2*n+1)*(2*n^2+2*n+45)/168): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011
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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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