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Search: id:A095875
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| A095875 |
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Number of lattice points on graph of parabola y >= x^2 with y <= n. |
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+0
1
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| 0, 1, 4, 7, 10, 15, 20, 25, 30, 35, 42, 49, 56, 63, 70, 77, 84, 93, 102, 111, 120, 129, 138, 147, 156, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 470, 485, 500, 515, 530, 545, 560
(list; graph; listen)
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OFFSET
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-1,3
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COMMENT
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Positive terms are partial sums of A001650, n appears n times (n odd).
a(n) is typically larger than the analytical integral (4/3)n^(3/2) of the area because integer points right on the contour contribute with too much statistical weight in the Monte Carlo sense of area estimation. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2006
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FORMULA
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a(n+1)=a(n)+1+2*floor(sqrt(n+1))=a(n)+A001650(n+2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2006
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EXAMPLE
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a(2) = 7 because there are exactly seven points with integer coordinates within the graph of y >= x^2 and bounded by the line y = 2: (0,0), (-1,1), (0,1), (1,1), (-1,2), (0,2) and (1,2).
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MAPLE
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A095875 := proc(n) local y; sum(1+2*floor(sqrt(y)), y=0..n) ; end: for n from -1 to 60 do printf("%d, ", A095875(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2006
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PROGRAM
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(PARI) a(n) = sum(k=0, n, 1+2*sqrtint(n)) for(n=-1, 100, print1(a(n), ", "))
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CROSSREFS
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Cf. A001650 (first differences), A095876 (lattice points of corresponding paraboloid).
Sequence in context: A026372 A014690 A126891 this_sequence A071408 A137461 A137379
Adjacent sequences: A095872 A095873 A095874 this_sequence A095876 A095877 A095878
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KEYWORD
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nonn
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AUTHOR
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Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 10 2004
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