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A278046 Let v = list of denominators of Farey series of order n (see A006843); a(n) = sum of products of adjacent terms of v.

%I A278046 #24 Nov 24 2016 19:38:17
%S A278046 1,4,18,44,124,186,424,636,1038,1378,2368,2852,4516,5510,7030,8734,
%T A278046 12542,14168,19526,22206,26658,30728,40342,44190,54590,61402,72328,
%U A278046 80196,99684,105644,129514,143162,161422,176926,201566,214538,255386,277160,307736,329096,384856,402412,466826,499166
%N A278046 Let v = list of denominators of Farey series of order n (see A006843); a(n) = sum of products of adjacent terms of v.
%C A278046 Note that the sum of the reciprocals of these products is 1.
%H A278046 J. Lehner and M. Newman, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa15/aa15114.pdf">Sums involving Farey fractions</a>, Acta Arithmetica 15.2 (1969): 181-187.
%e A278046 When n = 4, v = [1,4,3,2,3,4,1], so a(4) = 1*4 + 4*3 + 3*2 + 2*3 + 3*4 + 4*1 = 44.
%p A278046 Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
%p A278046 ans:=[];
%p A278046 for n from 1 to 50 do
%p A278046 t1:=denom(Farey(n));
%p A278046 t2:=add( t1[i]*t1[i+1],i=1..nops(t1)-1);
%p A278046 ans:=[op(ans),t2];
%p A278046 od:
%p A278046 ans;
%Y A278046 Cf. A006843, A005728, A240877.
%K A278046 nonn
%O A278046 1,2
%A A278046 _N. J. A. Sloane_, Nov 22 2016