cp's OEIS Frontend

A136515 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter 2n+1.

%I A136515 #10 Jul 30 2023 02:13:08
%S A136515 0,2,6,12,26,38,56,74,96,128,154,188,220,262,304,344,398,452,506,562,
%T A136515 616,686,754,824,894,976,1056,1134,1224,1308,1406,1500,1592,1694,1804,
%U A136515 1914,2026,2136,2258,2374,2504,2626,2756,2892,3022,3164,3300,3450,3600
%N A136515 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter 2n+1.
%C A136515 Number of unit square lattice cells inside two adjacent quadrants of origin centered circle of radius n+1/2.
%H A136515 G. C. Greubel, <a href="/A136515/b136515.txt">Table of n, a(n) for n = 0..1000</a>
%F A136515 a(n) = 2*Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
%F A136515 Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
%F A136515 a(n) = 2 * A136484(n).
%F A136515 a(n) = (1/2)*A136486(2*n+1).
%e A136515 a(2) = 6 because a circle centered at the origin and of radius 2.5 encloses (-2,1),(-1,1),(-1,2),(2,1),(1,1),(1,2) in the upper half plane.
%t A136515 Table[2*Sum[Floor[Sqrt[(n +1/2)^2 -k^2]], {k,n}], {n,0,100}]
%o A136515 (Magma)
%o A136515 A136515:= func< n | n eq 0 select 0 else 2*(&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
%o A136515 [A136515(n): n in [0..100]]; // _G. C. Greubel_, Jul 27 2023
%o A136515 (SageMath)
%o A136515 def A136515(n): return 2*sum(isqrt((n+1/2)^2-k^2) for k in range(1,n+1))
%o A136515 [A136515(n) for n in range(101)] # _G. C. Greubel_, Jul 27 2023
%o A136515 (PARI) a(n) = 2*sum(k=1, n, sqrtint((n+1/2)^2-k^2)); \\ _Michel Marcus_, Jul 27 2023
%Y A136515 Cf. A136484, A136513, A136514.
%K A136515 easy,nonn
%O A136515 0,2
%A A136515 Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008