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A358298 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of lines defining the Farey diagram Farey(n,k) of order (n,k).

%I A358298 #34 Apr 03 2023 09:31:15
%S A358298 2,3,3,4,6,4,6,11,11,6,8,19,20,19,8,12,29,36,36,29,12,14,43,52,60,52,
%T A358298 43,14,20,57,78,88,88,78,57,20,24,77,100,128,124,128,100,77,24,30,97,
%U A358298 136,162,180,180,162,136,97,30,34,121,166,216,224,252,224,216,166,121,34
%N A358298 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of lines defining the Farey diagram Farey(n,k) of order (n,k).
%C A358298 We work with lines with equation ux + vy + w = 0 in the (x,y) plane.
%C A358298 This line has slope -u/v, and crosses the vertical y axis at the intercept point y = -w/v
%C A358298 For the Farey diagram Farey(m,n), u is an integer between -(m-1) and +(m-1), v is between -(n-1) and +(n-1) and w can be any integer.
%C A358298 The only lines that are used are those that hit the unit square 0 <= x <= 1, 0 <= y <= 1 in at least two points.
%C A358298 This means that we only need to look at w's with |w| <=  |u| + |v|.
%C A358298 T(m,n) is the number of such lines.
%C A358298 For illustrations of Farey(3,3) and Farey(3,4) see Khoshnoudirad (2015), Fig. 2, and Darat et al. (2009), Fig. 2. For further illustrations see A358882-A358885.
%H A358298 Alain Daurat, M. Tajine, and M. Zouaoui, <a href="https://doi.org/10.1016/j.cag.2008.11.001">About the frequencies of some patterns in digital planes. Application to area estimators</a>. Computers & Graphics. 33.1 (2009), 11-20.
%H A358298 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 A358298 The full array T(n,k), n >= 0, k>= 0, begins:
%e A358298   2, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60,  ...
%e A358298   3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205,  ...
%e A358298   4, 11, 20, 36, 52, 78, 100, 136, 166, 210, 246, 302,  ...
%e A358298   6, 19, 36, 60, 88, 128, 162, 216, 266, 326, 386, 468, ...
%e A358298   8, 29, 52, 88, 124, 180, 224, 298, 360, 444, 518, 628, ...
%e A358298   12, 43, 78, 128, 180, 252, 316, 412, 498, 608, 706,  ...
%e A358298   14, 57, 100, 162, 224, 316, 388, 508, 608, 738, 852, ...
%e A358298   ...
%p A358298 A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
%p A358298 Amn:=proc(m,n) local a,i,j;  # A331781 or equally A333295. Diagonal is A018805.
%p A358298 a:=0; for i from 1 to m do for j from 1 to n do
%p A358298 if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
%p A358298 # The present sequence is:
%p A358298 Dmn:=proc(m,n) local d,t1,u,v,a; global A005728, Amn;
%p A358298 a:=A005728(m)+A005728(n);
%p A358298 t1:=0; for u from 1 to m do for v from 1 to n do
%p A358298 d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
%p A358298 a+2*t1-2*Amn(m,n); end;
%p A358298 for m from 1 to 8 do lprint([seq(Dmn(m,n),n=1..20)]); od:
%t A358298 A005728[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
%t A358298 Amn[m_, n_] := Module[{a, i, j}, a = 0; For[i = 1, i <= m, i++, For[j = 1, j <= n, j++, If[GCD[i, j] == 1, a = a + 1]]]; a];
%t A358298 Dmn[m_, n_] := Module[{d, t1, u, v, a}, a = A005728[m] + A005728[n]; t1 = 0; For[u = 1, u <= m, u++, For[v = 1, v <= n, v++, d = GCD[u, v]; If[d >= 1 , t1 = t1 + (u + v)* EulerPhi[d]/d]]]; a + 2*t1 - 2*Amn[m, n]];
%t A358298 Table[Dmn[m - n, n], {m, 0, 10}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Apr 03 2023, after Maple code *)
%Y A358298 Cf. A358299.
%Y A358298 Row 0 is essentially A225531, row 1 is A358300, main diagonal is A358301.
%Y 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.
%K A358298 nonn,tabl
%O A358298 0,1
%A A358298 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 06 2022