cp's OEIS Frontend

A136514 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of radius n.

%I A136514 #11 Jul 30 2023 02:14:33
%S A136514 0,2,8,16,30,44,60,82,108,138,166,196,238,278,324,366,416,468,526,588,
%T A136514 644,714,780,848,930,1008,1090,1170,1256,1350,1438,1540,1638,1744,
%U A136514 1856,1954,2072,2180,2310,2432,2548,2678,2808,2950,3090,3220,3366,3510,3664
%N A136514 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of radius n.
%H A136514 G. C. Greubel, <a href="/A136514/b136514.txt">Table of n, a(n) for n = 1..1000</a>
%F A136514 Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
%F A136514 a(n) = 2 * Sum_{k=1..n-1} floor(sqrt(n^2 - k^2)).
%F A136514 a(n) = A136513(2*n).
%F A136514 a(n) = 2*A001182(n). - _R. J. Mathar_, Jan 10 2008
%e A136514 a(2) = 2 because a circle centered at the origin and of radius 2 encloses (-1,1) and (1,1) in the upper half plane.
%t A136514 Table[2*Sum[Floor[Sqrt[n^2 -k^2]], {k,n-1}], {n,100}]
%o A136514 (Magma)
%o A136514 A136514:= func< n | n eq 1 select 0 else 2*(&+[Floor(Sqrt(n^2-j^2)): j in [1..n-1]]) >;
%o A136514 [A136514(n): n in [1..100]]; // _G. C. Greubel_, Jul 27 2023
%o A136514 (SageMath)
%o A136514 def A136514(n): return 2*sum(isqrt(n^2-k^2) for k in range(1,n))
%o A136514 [A136514(n) for n in range(1,101)] # _G. C. Greubel_, Jul 27 2023
%o A136514 (PARI) a(n) = 2*sum(k=1, n-1, sqrtint(n^2-k^2)); \\ _Michel Marcus_, Jul 27 2023
%Y A136514 Cf. A001182, A136513.
%K A136514 easy,nonn
%O A136514 1,2
%A A136514 Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008