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A136486 Number of unit square lattice cells enclosed by origin centered circle of diameter 2n+1.

%I A136486 #6 Jul 31 2023 02:59:33
%S A136486 0,4,12,24,52,76,112,148,192,256,308,376,440,524,608,688,796,904,1012,
%T A136486 1124,1232,1372,1508,1648,1788,1952,2112,2268,2448,2616,2812,3000,
%U A136486 3184,3388,3608,3828,4052,4272,4516,4748,5008,5252,5512,5784,6044,6328,6600
%N A136486 Number of unit square lattice cells enclosed by origin centered circle of diameter 2n+1.
%C A136486 a(n) is the number of complete squares that fit inside the circle with radius n+1/2, drawn on squared paper.
%H A136486 G. C. Greubel, <a href="/A136486/b136486.txt">Table of n, a(n) for n = 0..1000</a>
%F A136486 a(n) = 4*Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
%F A136486 a(n) = 4 * A136484(n).
%F A136486 a(n) = 2 * A136515(n).
%F A136486 a(n) = A136485(2*n+1).
%F A136486 Lim_{n -> oo} a(n)/(n^2) -> Pi/4 (A003881).
%e A136486 a(1) = 4 because a circle centered at the origin and of radius 1+1/2 encloses (-1,-1), (-1,1), (1,-1), (1,1).
%t A136486 Table[4*Sum[Floor[Sqrt[(n + 1/2)^2 - k^2]], {k,n}], {n, 0, 100}]
%o A136486 (Magma)
%o A136486 A136486:= func< n | n eq 0 select 0 else 4*(&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
%o A136486 [A136486(n): n in [0..100]]; // _G. C. Greubel_, Jul 30 2023
%o A136486 (SageMath)
%o A136486 def A136486(n): return 4*sum(floor(sqrt((n+1/2)^2-k^2)) for k in range(1, n+1))
%o A136486 [A136486(n) for n in range(101)] # _G. C. Greubel_, Jul 30 2023
%Y A136486 Cf. A003881, A136484, A136485, A136515.
%K A136486 easy,nonn
%O A136486 0,2
%A A136486 Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008