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A358299 Triangle read by antidiagonals: T(n,k) (n>=0, 0 <= k <= n) = number of lines defining the Farey diagram of order (n,k).

%I A358299 #19 Dec 06 2022 19:33:37
%S A358299 2,3,6,4,11,20,6,19,36,60,8,29,52,88,124,12,43,78,128,180,252,14,57,
%T A358299 100,162,224,316,388,20,77,136,216,298,412,508,652,24,97,166,266,360,
%U A358299 498,608,780,924,30,121,210,326,444,608,738,940,1116,1332,34,145,246,386,518,706,852,1086,1280,1532,1748
%N A358299 Triangle read by antidiagonals: T(n,k) (n>=0, 0 <= k <= n) = number of lines defining the Farey diagram of order (n,k).
%H A358299 Daniel Khoshnoudirad, <a href="http://www.doiserbia.nb.rs/img/doi/1452-8630/2015/1452-86301500008K.pdf">Farey lines defining Farey diagrams and application to some discrete structures</a>, Applicable Analysis and Discrete Mathematics, 9 (2015), 73-84; doi:10.2298/AADM150219008K. See Theorem 1, |DF(m,n)|.
%e A358299 The full array T(n,k), n >= 0, k>= 0, begins:
%e A358299 2, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60, ...
%e A358299 3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205, ...
%e A358299 4, 11, 20, 36, 52, 78, 100, 136, 166, 210, 246, 302, ...
%e A358299 6, 19, 36, 60, 88, 128, 162, 216, 266, 326, 386, 468, ...
%e A358299 8, 29, 52, 88, 124, 180, 224, 298, 360, 444, 518, 628, ...
%e A358299 12, 43, 78, 128, 180, 252, 316, 412, 498, 608, 706, ...
%e A358299 14, 57, 100, 162, 224, 316, 388, 508, 608, 738, 852, ...
%e A358299 ...
%p A358299 A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
%p A358299 Amn:=proc(m,n) local a,i,j; # A331781 or equally A333295. Diagonal is A018805.
%p A358299 a:=0; for i from 1 to m do for j from 1 to n do
%p A358299 if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
%p A358299 # The present sequence is:
%p A358299 Dmn:=proc(m,n) local d,t1,u,v,a; global A005728, Amn;
%p A358299 a:=A005728(m)+A005728(n);
%p A358299 t1:=0; for u from 1 to m do for v from 1 to n do
%p A358299 d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
%p A358299 a+2*t1-2*Amn(m,n); end;
%p A358299 for m from 1 to 8 do lprint([seq(Dmn(m,n),n=1..20)]); od:
%Y A358299 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.
%K A358299 nonn,tabl
%O A358299 0,1
%A A358299 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 06 2022