A358305 Triangle read by rows: T(n,k) (n>=0, 0 <= k <= n) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

0, 0, 2, 0, 5, 10, 0, 9, 19, 32, 0, 14, 27, 47, 66, 0, 20, 40, 68, 96, 134, 0, 27, 51, 85, 118, 167, 204, 0, 35, 68, 112, 156, 217, 267, 342, 0, 44, 82, 137, 187, 261, 318, 408, 482, 0, 54, 103, 166, 229, 317, 384, 490, 581, 692, 0, 65, 120, 196, 266, 366, 441, 564, 664, 794, 904

Offset

0,3

Links

Table of n, a(n) for n=0..65.

Daniel Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures, Applicable Analysis and Discrete Mathematics, 9 (2015), 73-84; doi:10.2298/AADM150219008K. See Lemma 1, |DFD(m,n)|.

Example

The full array T(n,k), n >= 0, k>= 0, begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, ...
0, 5, 10, 19, 27, 40, 51, 68, 82, 103, 120, 145, ...
0, 9, 19, 32, 47, 68, 85, 112, 137, 166, 196, 235, ...
0, 14, 27, 47, 66, 96, 118, 156, 187, 229, 266, ...
0, 20, 40, 68, 96, 134, 167, 217, 261, 317, 366, ...
0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, ...

Maple

A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
Amn:=proc(m, n) local a, i, j; # A331781 or equally A333295. Diagonal is A018805.
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;
DFD:=proc(m, n) local d, t1, u, v; 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:
t1; end;
for m from 0 to 8 do lprint([seq(DFD(m, n), n=0..20)]); od:

Mathematica

T[n_, k_] := Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, n}, {v, 1, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2023 *)

Crossrefs

Cf. A358298.

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Sequence in context: A344786 A011014 A002976 * A292590 A080901 A137260

Adjacent sequences: A358302 A358303 A358304 * A358306 A358307 A358308

Keyword

nonn,tabl

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Status

approved