|
| |
|
|
A085582
|
|
The number of rectangles (orthogonal or not) with corners on an n X n grid of points.
|
|
10
|
|
|
|
0, 1, 10, 44, 130, 313, 640, 1192, 2044, 3305, 5078, 7524, 10750, 14993, 20388, 27128, 35448, 45665, 57922, 72636, 89970, 110297, 133976, 161440, 192860, 228857, 269758, 316012, 367974, 426417, 491468, 564120, 644640, 733633, 831674, 939292
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
LINKS
|
David Radcliffe, Table of n, a(n) for n = 1..1000
David Radcliffe, Python script to calculate a(n)
|
|
|
FORMULA
|
a(n) = A000537(n-1) + A113751(n). - T. D. Noe, Nov 09 2005 [corrected by David Radcliffe, Feb 06 2020]
a(n) = n*(n-1)^2*(2n-1)/6 + 2*Sum_{a,b>0, 0<s<r<n, gcd(r,s)=1} max(n-a*s-b*r,0)*max(n-a*r-b*s,0). - David Radcliffe, Feb 06 2020
|
|
|
EXAMPLE
|
a(3) = 10 because on the 3 X 3 grid there are four 1 X 1 rectangles, two 1 X 2s, two 2 X 1's, one 2 X 2 and one 45-degree rectangle, sqrt(2) X sqrt(2).
|
|
|
CROSSREFS
|
Cf. A000537, A002415, A113751 (diagonal rectangles on an n X n grid).
Sequence in context: A256050 A257052 A008532 * A058310 A005720 A060326
Adjacent sequences: A085579 A085580 A085581 * A085583 A085584 A085585
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 06 2003
|
|
|
EXTENSIONS
|
Edited by Don Reble, Nov 05 2005
|
|
|
STATUS
|
approved
|
| |
|
|
