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A175869
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Partial sums of ceiling(n^2/23).
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1
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0, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 44, 53, 63, 75, 88, 103, 119, 137, 157, 179, 202, 228, 256, 286, 318, 353, 390, 430, 472, 517, 565, 616, 670, 727, 787, 850, 917, 987, 1061, 1138, 1219, 1304, 1393, 1485, 1582, 1683, 1788, 1897
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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,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).
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FORMULA
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a(n) = round((2*n+1)*(n^2 + n + 42)/138).
a(n) = floor((2*n^3 + 3*n^2 + 85*n + 90)/138).
a(n) = ceiling((2*n^3 + 3*n^2 + 85*n - 6)/138).
a(n) = a(n-23) + (n+1)*(n-23) + 202.
G.f.: x*(x + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^6 - x^3 + 1)*(x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x - 1)^4*(x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + 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(23) = 0 + 1 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 12 + 13 + 15 + 16 + 18 + 20 + 22 + 23 = 202.
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MAPLE
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seq(round((2*n+1)*(n^2+n+42)/138), n=0..50)
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MATHEMATICA
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Accumulate[Ceiling[Range[0, 50]^2/23]] (* Harvey P. Dale, Feb 04 2019 *)
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PROG
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(Magma) [Floor((2*n^3+3*n^2+85*n+90)/138): n in [0..50]]; // Vincenzo Librandi, Apr 29 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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