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 A112543 Numerators of fractions n/m in array by antidiagonals. 3

%I

%S 1,2,1,3,1,1,4,3,2,1,5,2,1,1,1,6,5,4,3,2,1,7,3,5,1,3,1,1,8,7,2,5,4,1,

%T 2,1,9,4,7,3,1,2,3,1,1,10,9,8,7,6,5,4,3,2,1,11,5,3,2,7,1,5,1,1,1,1,12,

%U 11,10,9,8,7,6,5,4,3,2,1,13,6,11,5,9,4,1,3,5,2,3,1,1,14,13,4,11,2,3,8,7,2

%N Numerators of fractions n/m in array by antidiagonals.

%C Column m has period m. - _Clark Kimberling_, Jul 04 2013

%C Read as a triangle with antidiagonals for rows, T(n,k) gives the number of distinct locations at which two points the same distance from a center rotating around that center in the same direction at speeds n+1 and k will coincide. - _Thomas Anton_, Nov 23 2018

%H Clark Kimberling, <a href="/A112543/b112543.txt">Antidiagonals n = 1..60, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>

%F For n/m: g.f.: x/(1-x)*log(1/(1-y)), e.g.f.: x*e^x*(Ei(y)-log(y)+gamma) = x*e^x*integral_t=0^y (e^t-1)dt.

%F n/gcd(m,n). - _Clark Kimberling_, Jul 04 2013

%e a(2,4)=1/2 because 2/4 = 1/2.

%e Northwest corner:

%e 1 2 3 4 5 6 7

%e 1 1 3 2 5 3 7

%e 1 2 1 4 5 2 7

%e 1 1 3 1 5 3 7

%e 1 2 3 4 1 1 7

%e 1 1 1 2 5 1 7

%e 1 2 3 4 5 6 1

%t d[m_, n_] := n/GCD[m, n]; z = 12;

%t TableForm[Table[d[m, n], {m, 1, z}, {n, 1, z}] ] (*array*)

%t Flatten[Table[d[k, m + 1 - k], {m, 1, z}, {k, 1, m}]] (*sequence*)

%t (* _Clark Kimberling_, Jul 04 2013 *)

%o (PARI) t1(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) vector(100,n,t1(n)/gcd(t1(n),t2(n)))

%Y Denominators in A112544. Reduced version of A004736/A002260.

%K easy,frac,nonn,tabl

%O 1,2

%A _Franklin T. Adams-Watters_, Sep 11 2005