A358304 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

0, 0, 0, 0, 2, 0, 0, 5, 5, 0, 0, 9, 10, 9, 0, 0, 14, 19, 19, 14, 0, 0, 20, 27, 32, 27, 20, 0, 0, 27, 40, 47, 47, 40, 27, 0, 0, 35, 51, 68, 66, 68, 51, 35, 0, 0, 44, 68, 85, 96, 96, 85, 68, 44, 0, 0, 54, 82, 112, 118, 134, 118, 112, 82, 54, 0, 0, 65, 103, 137, 156, 167, 167, 156, 137, 103, 65, 0, 0, 77, 120, 166, 187, 217, 204, 217, 187, 166, 120, 77, 0

Offset

0,5

Links

Table of n, a(n) for n=0..90.

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,  2,  5,  9,  14,  20,  27,  35,  44,  54,  65,  77,  90, ..
  0,  5, 10, 19,  27,  40,  51,  68,  82, 103, 120, 145, 165, ..
  0,  9, 19, 32,  47,  68,  85, 112, 137, 166, 196, 235, 265, ..
  0, 14, 27, 47,  66,  96, 118, 156, 187, 229, 266, 320, 358, ..
  0, 20, 40, 68,  96, 134, 167, 217, 261, 317, 366, 436, 491, ..
  0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, 528, 589, ..
  ...

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, k], {n, 0, 12}, {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: A340116 A361521 A223705 * A285192 A278177 A095221

Adjacent sequences: A358301 A358302 A358303 * A358305 A358306 A358307

Keyword

nonn,tabl

Author

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

Status

approved