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

Author

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

Keywords

• nonn
• tabl

Examples

			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, ..
  ...
		

Links

• 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)|.

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.

Programs

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 *)