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A333168 a(n) = Sum_{k=0..n} r_2(k^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).

%I A333168 #9 Mar 10 2020 02:47:42
%S A333168 4,8,16,24,32,40,48,60,76,84,92,100,116,132,140,148,156,172,196,204,
%T A333168 212,228,244,260,268,276,284,300,316,324,340,356,380,396,412,420,428,
%U A333168 444,468,476,484,496,512,536,552,560,576,608,624,632,648,656,672,688,696
%N A333168 a(n) = Sum_{k=0..n} r_2(k^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).
%D A333168 Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.
%H A333168 Amiram Eldar, <a href="/A333168/b333168.txt">Table of n, a(n) for n = 0..10000</a>
%H A333168 E. J. Scourfield, <a href="https://doi.org/10.1017/S2040618500034237">The divisors of a quadratic polynomial</a>, Glasgow Mathematical Journal, Vol. 5, No. 1 (1961), pp. 8-20.
%F A333168 a(n) ~ (8/Pi) * n * log(n).
%e A333168 a(0) = r_2(0^2 + 1) = r_2(1) = A004018(1) = 4.
%e A333168 a(1) = r_2(0^2 + 1) + r_2(1^1 + 1) = r_2(1) + r_2(2) = A004018(1) + A004018(2) = 4 + 4 = 8.
%t A333168 Accumulate @ Table[SquaresR[2, k^2 + 1], {k, 0, 100}]
%Y A333168 Partial sums of A333167.
%Y A333168 Cf. A002522, A004018.
%K A333168 nonn
%O A333168 0,1
%A A333168 _Amiram Eldar_, Mar 09 2020