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Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.
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%I #36 Jan 15 2022 00:51:42

%S 0,1,2,2,3,5,5,5,6,7,9,9,9,11,11,11,12,14,15,15,17,17,17,17,17,20,22,

%T 22,22,24,24,24,25,25,27,27,28,30,30,30,32,34,34,34,34,36,36,36,36,37,

%U 40,40,42,44,44,44,44,44,46

%N Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.

%C From _Ant King_, Mar 15 2013: (Start)

%C The terms of this sequence give a running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors.

%C To see how good the approximation n * Pi/4 is to a(n), note that a(10^6) = 785387 whereas 10^6 * Pi/4 rounds to 785398. (End)

%H Seiichi Manyama, <a href="/A014200/b014200.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A014198(n) / 4.

%F Limit_{n->infinity} a(n)/n = Pi/4.

%F a(n) = n - floor(n/3) + floor(n/5) - floor(n/7) + floor(n/9) - floor(n/11) + ... - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

%F G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 + x^(2*k)). - _Ilya Gutkovskiy_, Dec 23 2016

%t 1/4*Prepend[SquaresR[2,#]&/@Range[58],0]//Accumulate (* _Ant King_, Mar 15 2013 *)

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d))); \\ _Seiichi Manyama_, Dec 18 2021

%Y Cf. A014198, A059851, A101455.

%Y Partial sums of A002654.

%K nonn

%O 0,3

%A _N. J. A. Sloane_