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A177237
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Partial sums of round(n^2/19).
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1
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0, 0, 0, 0, 1, 2, 4, 7, 10, 14, 19, 25, 33, 42, 52, 64, 77, 92, 109, 128, 149, 172, 197, 225, 255, 288, 324, 362, 403, 447, 494, 545, 599, 656, 717, 781, 849, 921, 997, 1077, 1161, 1249, 1342, 1439, 1541, 1648, 1759, 1875, 1996, 2122, 2254
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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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Vincenzo Librandi, Table of n, a(n) for n = 0..895
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
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FORMULA
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a(n)=round((n-2)*(n+3)*(2*n+1)/114).
a(n)=floor((2*n^3+3*n^2-11*n+42)/114).
a(n)=ceil((2*n^3+3*n^2-11*n-54)/114).
a(n)=round((2*n^3+3*n^2-11*n)/114).
a(n)=a(n-19)+(n+1)*(n-19)+128 , n>18.
a(n)= +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-19) -3*a(n-20) +3*a(n-21) -a(n-22) with G.f. x^4*(1+x)*(x^4-x^3+x^2-x+1)*(x^8-x^7+x^6-x^4+x^2-x+1) / ( (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) *(x-1)^4 ). [R. J. Mathar, Dec 13 2010]
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EXAMPLE
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a(19) = 0+0+0+0+1+1+2+3+3+4+5+6+8+9+10+12+13+15+17+19 = 128.
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MAPLE
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seq(round((2*n^3+3*n^2-11*n)/114), n=0..50)
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PROG
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(MAGMA) [Floor((2*n^3+3*n^2-11*n+42)/114): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011
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CROSSREFS
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Cf. A177100, A177116.
Sequence in context: A023536 A196126 A024536 * A094281 A076101 A170890
Adjacent sequences: A177234 A177235 A177236 * A177238 A177239 A177240
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KEYWORD
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nonn,easy
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AUTHOR
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Mircea Merca, Dec 10 2010
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STATUS
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approved
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