A014200 Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.

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, 22, 22, 24, 24, 24, 25, 25, 27, 27, 28, 30, 30, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 37, 40, 40, 42, 44, 44, 44, 44, 44, 46

Offset

0,3

Comments

From Ant King, Mar 15 2013: (Start)

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.

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)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

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

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

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

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

Mathematica

1/4*Prepend[SquaresR[2, #]&/@Range[58], 0]//Accumulate (* Ant King, Mar 15 2013 *)

Prog

(PARI) a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d))); \\ Seiichi Manyama, Dec 18 2021

Crossrefs

Cf. A014198, A059851, A101455.

Partial sums of A002654.

Sequence in context: A055769 A162217 A123575 * A293522 A319476 A140200

Adjacent sequences: A014197 A014198 A014199 * A014201 A014202 A014203

Keyword

nonn

Author

N. J. A. Sloane

Status

approved