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A014198 Number of integer solutions to x^2 + y^2 <= n excluding (0,0). 5
0, 4, 8, 8, 12, 20, 20, 20, 24, 28, 36, 36, 36, 44, 44, 44, 48, 56, 60, 60, 68, 68, 68, 68, 68, 80, 88, 88, 88, 96, 96, 96, 100, 100, 108, 108, 112, 120, 120, 120, 128, 136, 136, 136, 136, 144, 144, 144, 144, 148, 160, 160, 168, 176, 176, 176, 176, 176, 184, 184 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(32)/32 = 100/32 = 3.125; lim_{n->inf} a(n)/n = Pi.

The terms of this sequence are four times the running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors. - Ant King, Mar 12 2013

REFERENCES

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 339

LINKS

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

Eric Weisstein's World of Mathematics, Sum of Squares Function

FORMULA

a(n) = 4*A014200(n).

a(n) = A057655(n)-1.

EXAMPLE

For n=2 the 8 solutions are (x,y) = (+-1,0), (0,+-1), (+-1,+-1).

MAPLE

A014198 := proc(n)

    nops([ numtheory[thue]( abs( x^2+y^2) <= n, [ x, y ] ) ]);

end proc:

seq(A014198(n), n=0..60) ;

MATHEMATICA

Prepend[SquaresR[2, #] &/@Range[59], 0]//Accumulate (* Ant King, Mar 12 2013 *)

PROG

(PARI) a(n)=local(j); j=sqrtint(n); sum(x=-j, j, sum(y=-j, j, x^2+y^2<=n))-1

CROSSREFS

Cf. A014200, A057655.

Sequence in context: A273456 A299771 A294963 * A316316 A159786 A083744

Adjacent sequences:  A014195 A014196 A014197 * A014199 A014200 A014201

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 22 01:33 EST 2019. Contains 320381 sequences. (Running on oeis4.)