A228108 Number of pairs (x, y) with 0 <= x, y <= n such that the distance between two points is a positive integer.

4, 18, 48, 108, 204, 342, 528, 780, 1100, 1494, 1968, 2576, 3292, 4122, 5104, 6240, 7524, 8962, 10560, 12348, 14324, 16494, 18864, 21600, 24572, 27786, 31248, 34996, 39012, 43362, 48000, 52968, 58244, 63834, 69840, 76308, 83132, 90318, 97872, 105972, 114468, 123378, 132704, 142500, 152892, 163742, 175056

Offset: 1

Author

David A. Corneth, Aug 10 2013

Keywords

• nonn

Comments

Pairs ((0, 1), (0, 0)) and ((0, 0), (0, 1)) are considered equal since they have the same points. ((0, 0), (0, 0)) isn't counted the distance between the points included isn't a positive integer. The x's and y's of points in a pair may differ due to Pythagoras, for example ((0, 0), (3, 4)).

Links

• David A. Corneth, Table of n, a(n) for n = 1..10560
• Wikipedia, Pythagorean triple

Programs

PARI

a(n)=my(tot=n*(n+1)^2);forstep(i=1,sqrt(ceil(sqrt(n))^2)*(1+sqrt(2))\1,2,forstep(j=2,max(sqrt(ceil(sqrt(n+i^2))^2)\1,n\(2*i))*(1+sqrt(2))\1,2,if(gcd(i,j)==1,for(k=1,n\max(2*i*j,abs(j^2-i^2)),
tot+=(n-2*i*j*k+1)*(n-abs(j^2-i^2)*k+1)*4)))); tot \\David A. Corneth