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

%I #23 Jul 17 2024 09:21:39

%S 0,1,4,7,10,15,20,25,30,35,42,49,56,63,70,77,84,93,102,111,120,129,

%T 138,147,156,165,176,187,198,209,220,231,242,253,264,275,286,299,312,

%U 325,338,351,364,377,390,403,416,429,442,455,470,485,500,515,530,545,560

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

%C Positive terms are partial sums of A001650, n appears n times (n odd).

%C 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

%F a(n+1) = a(n)+1+2*floor(sqrt(n+1)) = a(n)+A001650(n+2). - _R. J. Mathar_, Nov 06 2006

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

%e 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).

%p 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

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

%o (PARI) a(n) = sum(k=0, n, 1+2*sqrtint(k)); \\ corrected by _Michel Marcus_, Feb 07 2023

%o for(n=-1,100,print1(a(n),", "))

%Y Cf. A001650 (first differences).

%K nonn

%O -1,3

%A _Rick L. Shepherd_, Jun 10 2004