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A112543
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Numerators of fractions n/k in array by antidiagonals.
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3
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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, 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, 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, 1, 4, 1, 2, 1
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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For n/k: (Start)
G.f.: x/(1-x)*log(1/(1-y)),
E.g.f.: x*exp(x)*(Ei(y) - log(y) + EulerGamma) = x*e^x*Integral_{t=0}^{y} (exp(t) - 1) dt. (End)
A(n, k) = numerator(k/n) (array).
T(n, k) = numerator((n-k+1)/k) (antidiagonals).
Sum_{k=1..n} T(n, k) = A332049(n+1).
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EXAMPLE
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a(2,4) = 1/2 because 2/4 = 1/2.
Northwest corner of the array:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
1, 1, 3, 2, 5, 3, 7, 4, 9, 5, ...;
1, 2, 1, 4, 5, 2, 7, 8, 3, 10, ...;
1, 1, 3, 1, 5, 3, 7, 2, 9, 5, ...;
1, 2, 3, 4, 1, 6, 7, 8, 9, 2, ...;
1, 1, 1, 2, 5, 1, 7, 4, 3, 5, ...;
1, 2, 3, 4, 5, 6, 1, 8, 9, 10, ...;
1, 1, 3, 1, 5, 3, 7, 1, 9, 5, ...;
1, 2, 1, 4, 5, 2, 7, 8, 1, 10, ...;
1, 1, 3, 2, 1, 3, 7, 4, 9, 1, ...;
Antidiagonal triangle begins:
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, 2, 1;
9, 4, 7, 3, 1, 2, 3, 1, 1;
10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
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MATHEMATICA
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d[m_, n_] := n/GCD[m, n]; z = 12;
TableForm[Table[d[m, n], {m, 1, z}, {n, 1, z}] ] (*array*)
Flatten[Table[d[k, m + 1 - k], {m, 1, z}, {k, 1, m}]] (*sequence*)
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PROG
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(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)))
(Magma) [Numerator((n-k+1)/k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jan 12 2022
(Sage) flatten([[numerator((n-k+1)/k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Jan 12 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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