A333174 - OEIS

%I #8 Mar 10 2020 02:48:02

%S 8,32,80,224,368,704,1008,1752,2424,3432,4248,5736,7176,9768,11352,

%T 14088,16152,20472,23944,28312,31528,37576,42280,50056,54680,62216,

%U 67640,78296,85880,96008,103784,116552,126968,142808,152888,167624,178008,197880,212616,230904

%N a(n) = Sum_{k=0..n} r_4(k^2 + 1), where r_4(k) is the number of ways of writing k as a sum of 4 squares (A000118).

%H Amiram Eldar, <a href="/A333174/b333174.txt">Table of n, a(n) for n = 0..10000</a>

%H R. Sitaramachandrarao and P. V. Krishnaiah, <a href="https://doi.org/10.1016/0022-314X(86)90086-7">On the sums Sigma_{n<=x} A(f(n)) and Sigma_{p<=x} A(f(p))</a>, Journal of Number Theory, Vol. 23, No. 2 (1986), pp. 149-168.

%F a(n) ~ (40*G/Pi^2) * n^3, where G is Catalan's constant (A006752).

%e a(0) = r_4(0^2 + 1) = r_4(1) = A000118(1) = 8.

%e a(1) = r_4(0^2 + 1) + r_4(1^1 + 1) = r_4(1) + r_4(2) = A000118(1) + A000118(2) = 8 + 24 = 32.

%t Accumulate @ Table[SquaresR[4, k^2 + 1], {k, 0, 100}]

%Y Partial sums of A333173.

%Y Cf. A000118, A002522, A006752.

%K nonn

%O 0,1

%A _Amiram Eldar_, Mar 09 2020