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A177332
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Partial sums of round(n^2/29).
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1
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0, 0, 0, 0, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 35, 43, 52, 62, 73, 85, 99, 114, 131, 149, 169, 191, 214, 239, 266, 295, 326, 359, 394, 432, 472, 514, 559, 606, 656, 708, 763, 821, 882, 946, 1013, 1083, 1156, 1232, 1311, 1394, 1480
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OFFSET
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0,6
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COMMENTS
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The round function is defined here by round(x) = floor(x + 1/2).
There are several sequences of integers of the form round(n^2/k) for whose partial sums we can establish identities as following (only for k = 2, ..., 9, 11, 12, 13, 16, 17, 19, 20, 28, 29, 36, 44).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).
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FORMULA
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a(n) = round(n*(n+1)*(2*n+1)/174).
a(n) = floor((n+4)*(2*n^2 - 5*n + 21)/174).
a(n) = ceiling((n-3)*(2*n^2 + 9*n + 28)/174).
a(n) = a(n-29) + (n+1)*(n-29) + 266, n > 28.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-29) - 3*a(n-30) + 3*a(n-31) - a(n-32). - R. J. Mathar, Dec 13 2010
G.f.: x^4*(x+1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^10 - x^6 + x^5 - x^4 + 1)/((x-1)^4*(x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + 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, Apr 06 2012
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EXAMPLE
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a(17) = 0 + 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 62.
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MAPLE
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seq(round(n*(n+1)*(2*n+1)/174), n=0..50)
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MATHEMATICA
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Accumulate[Table[Round[n^2/29], {n, 0, 60}]] (* Harvey P. Dale, Dec 18 2010 *)
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PROG
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(Magma) [Floor((n+4)*(2*n^2-5*n+21)/174): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011
(Python)
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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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