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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). 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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 Theorem 1, |DF(m,n)|. EXAMPLE 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, ... ... 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: CROSSREFS 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: A049449 A351378 A104663 * A246429 A305325 A302848 Adjacent sequences: A358296 A358297 A358298 * A358300 A358301 A358302 KEYWORD nonn,tabl AUTHOR Scott R. Shannon and N. J. A. Sloane, Dec 06 2022 STATUS approved

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Last modified April 24 04:14 EDT 2024. Contains 371918 sequences. (Running on oeis4.)