A358299 Triangle read by antidiagonals: T(n,k) (n>=0, 0 <= k <= n) = number of lines defining the Farey diagram of order (n,k).
2, 3, 6, 4, 11, 20, 6, 19, 36, 60, 8, 29, 52, 88, 124, 12, 43, 78, 128, 180, 252, 14, 57, 100, 162, 224, 316, 388, 20, 77, 136, 216, 298, 412, 508, 652, 24, 97, 166, 266, 360, 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
Offset: 0
Examples
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, ... ...
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 Theorem 1, |DF(m,n)|.
Crossrefs
Programs
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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; # The present sequence is: Dmn:=proc(m,n) local d,t1,u,v,a; global A005728, Amn; a:=A005728(m)+A005728(n); 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: