A095875 Number of lattice points on graph of parabola y >= x^2 with y <= n.

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

Offset

-1,3

Comments

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, Nov 06 2006

Links

Table of n, a(n) for n=-1..55.

Formula

a(n+1) = a(n)+1+2*floor(sqrt(n+1)) = a(n)+A001650(n+2). - R. J. Mathar, Nov 06 2006

G.f.: theta_3(x)/(1 - x)^2, where theta_() is the Jacobi theta function. - Ilya Gutkovskiy, Jan 18 2018

Example

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).

Maple

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, Nov 06 2006

Mathematica

Join[{0}, Table[Array[k&, k], {k, 1, 15, 2}] // Flatten // Accumulate] (* Jean-François Alcover, Jul 17 2024 *)

Prog

(PARI) a(n) = sum(k=0, n, 1+2*sqrtint(k)); \\ corrected by Michel Marcus, Feb 07 2023
for(n=-1, 100, print1(a(n), ", "))

Crossrefs

Cf. A001650 (first differences).

Sequence in context: A126891 A310705 A310706 * A071408 A137461 A137379

Adjacent sequences: A095872 A095873 A095874 * A095876 A095877 A095878

Keyword

nonn

Author

Rick L. Shepherd, Jun 10 2004

Status

approved