|
| |
|
|
A358298
|
|
Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of lines defining the Farey diagram Farey(n,k) of order (n,k).
|
|
18
|
|
|
|
2, 3, 3, 4, 6, 4, 6, 11, 11, 6, 8, 19, 20, 19, 8, 12, 29, 36, 36, 29, 12, 14, 43, 52, 60, 52, 43, 14, 20, 57, 78, 88, 88, 78, 57, 20, 24, 77, 100, 128, 124, 128, 100, 77, 24, 30, 97, 136, 162, 180, 180, 162, 136, 97, 30, 34, 121, 166, 216, 224, 252, 224, 216, 166, 121, 34
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
We work with lines with equation ux + vy + w = 0 in the (x,y) plane.
This line has slope -u/v, and crosses the vertical y axis at the intercept point y = -w/v
For the Farey diagram Farey(m,n), u is an integer between -(m-1) and +(m-1), v is between -(n-1) and +(n-1) and w can be any integer.
The only lines that are used are those that hit the unit square 0 <= x <= 1, 0 <= y <= 1 in at least two points.
This means that we only need to look at w's with |w| <= |u| + |v|.
T(m,n) is the number of such lines.
For illustrations of Farey(3,3) and Farey(3,4) see Khoshnoudirad (2015), Fig. 2, and Darat et al. (2009), Fig. 2. For further illustrations see A358882-A358885.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
The full array T(n,k), n >= 0, k>= 0, begins:
2, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60, ...
3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205, ...
4, 11, 20, 36, 52, 78, 100, 136, 166, 210, 246, 302, ...
6, 19, 36, 60, 88, 128, 162, 216, 266, 326, 386, 468, ...
8, 29, 52, 88, 124, 180, 224, 298, 360, 444, 518, 628, ...
12, 43, 78, 128, 180, 252, 316, 412, 498, 608, 706, ...
14, 57, 100, 162, 224, 316, 388, 508, 608, 738, 852, ...
...
|
|
|
MAPLE
|
A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
a:=0; for i from 1 to m do for j from 1 to n do
if igcd(i, j)=1 then a:=a+1; fi; od: od: a; end;
# The present sequence is:
Dmn:=proc(m, n) local d, t1, u, v, a; global A005728, Amn;
t1:=0; for u from 1 to m do for v from 1 to n do
d:=igcd(u, v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
a+2*t1-2*Amn(m, n); end;
for m from 1 to 8 do lprint([seq(Dmn(m, n), n=1..20)]); od:
|
|
|
MATHEMATICA
|
A005728[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
Amn[m_, n_] := Module[{a, i, j}, a = 0; For[i = 1, i <= m, i++, For[j = 1, j <= n, j++, If[GCD[i, j] == 1, a = a + 1]]]; a];
Dmn[m_, n_] := Module[{d, t1, u, v, a}, a = A005728[m] + A005728[n]; t1 = 0; For[u = 1, u <= m, u++, For[v = 1, v <= n, v++, d = GCD[u, v]; If[d >= 1 , t1 = t1 + (u + v)* EulerPhi[d]/d]]]; a + 2*t1 - 2*Amn[m, n]];
Table[Dmn[m - n, n], {m, 0, 10}, {n, 0, m}] // Flatten (* Jean-François Alcover, Apr 03 2023, after Maple code *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
