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A136484 Number of unit square lattice cells inside quadrant of origin centered circle of diameter 2n+1.

%I A136484 #6 Jul 30 2023 02:14:27
%S A136484 0,1,3,6,13,19,28,37,48,64,77,94,110,131,152,172,199,226,253,281,308,
%T A136484 343,377,412,447,488,528,567,612,654,703,750,796,847,902,957,1013,
%U A136484 1068,1129,1187,1252,1313,1378,1446,1511,1582,1650,1725,1800,1877,1955,2034
%N A136484 Number of unit square lattice cells inside quadrant of origin centered circle of diameter 2n+1.
%C A136484 Number of unit square lattice cells inside quadrant of origin centered circle of radius n+1/2.
%H A136484 G. C. Greubel, <a href="/A136484/b136484.txt">Table of n, a(n) for n = 0..1000</a>
%F A136484 a(n) = Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
%F A136484 a(n) = (1/2) * A136515(n).
%F A136484 a(n) = (1/4) * A136486(n).
%F A136484 a(n) = A136483(2*n+1).
%F A136484 Lim_{n -> oo} a(n)/(n^2) -> Pi/16 (A019683).
%e A136484 a(2) = 3 because a circle of radius 2+1/2 in the first quadrant encloses (2,1), (1,1), (1,2).
%t A136484 Table[Sum[Floor[Sqrt[(n+1/2)^2 - k^2]], {k,n}], {n,0,100}]
%o A136484 (Magma)
%o A136484 A136484:= func< n | n eq 0 select 0 else (&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
%o A136484 [A136484(n): n in [0..100]]; // _G. C. Greubel_, Jul 29 2023
%o A136484 (SageMath)
%o A136484 def A136484(n): return sum(floor(sqrt((n+1/2)^2-k^2)) for k in range(1, n+1))
%o A136484 [A136484(n) for n in range(101)] # _G. C. Greubel_, Jul 29 2023
%Y A136484 Cf. A019683, A136483, A136486, A136515.
%K A136484 easy,nonn
%O A136484 0,3
%A A136484 Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008