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A177116
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Partial sums of round(n^2/11).
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10
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0, 0, 0, 1, 2, 4, 7, 11, 17, 24, 33, 44, 57, 72, 90, 110, 133, 159, 188, 221, 257, 297, 341, 389, 441, 498, 559, 625, 696, 772, 854, 941, 1034, 1133, 1238, 1349, 1467, 1591, 1722, 1860, 2005, 2158, 2318, 2486, 2662, 2846, 3038, 3239, 3448, 3666, 3893
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OFFSET
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0,5
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COMMENTS
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The round function, also called the nearest integer 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,-3,1,0,0,0,0,0,0,0,1,-3,3,-1).
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FORMULA
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a(n) = round((n-2)*(n+3)*(2*n+1)/66).
a(n) = floor((2*n^3 + 3*n^2 - 11*n + 18)/66).
a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 30)/66).
a(n) = round(n*(2*n^2 + 3*n - 11)/66).
a(n) = a(n-11) + (n+1)*(n-11) + 44, n > 10.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14). [R. J. Mathar, Dec 10 2010]
G.f.: x^3 *(1+x) *(x^2-x+1) *(x^4-x^3+x^2-x+1) / ( (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) *(x-1)^4 ). [R. J. Mathar, Dec 10 2010; Typo fixed by Colin Barker, Oct 10 2012
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EXAMPLE
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a(11) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 6 + 7 + 9 + 11 = 44.
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MAPLE
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seq(round((2*n^3+3*n^2-11*n)/66), n=0..50)
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MATHEMATICA
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Accumulate[Round[Range[0, 50]^2/11]] (* or *) LinearRecurrence[{3, -3, 1, 0, 0, 0, 0, 0, 0, 0, 1, -3, 3, -1}, {0, 0, 0, 1, 2, 4, 7, 11, 17, 24, 33, 44, 57, 72}, 60] (* Harvey P. Dale, Dec 10 2014 *)
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PROG
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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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