OFFSET
1,2
COMMENTS
Column k has period k. - Clark Kimberling, Jul 04 2013
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
As a rectangular array, each row sequence is multiplicative and strongly divisible, as discussed in Comments in A060819. The fractions, as n/gcd(m,n) are generated from an initial generation G(0) = (m/1) by applying the function f(x,y) = x*y/(x+y) = 1/(1/x + 1/y) to successive generations; i.e., for n>=1, G(n) = f(G(n-1)). Duplicates are removed as they occur, leaving 2^m fractions in G(m), for m>=0. See Example. - Clark Kimberling, Aug 04 2025
LINKS
Clark Kimberling, Antidiagonals n = 1..60, flattened
Eric Weisstein's World of Mathematics, Exponential Integral
FORMULA
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)
T(n, k) = n/gcd(k, n). - Clark Kimberling, Jul 04 2013
From G. C. Greubel, Jan 12 2022: (Start)
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).
T(n, k) = A112544(n, n-k). (End)
EXAMPLE
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;
Regarding the generations G(n) mentioned in Comments, for m = 2, the first 4 generations are G(0) = (2), G(1) = (2,1), G(2) = (2, 1, 2/3, 1/2), G(3) = (2, 1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4); the denominators in G(3) are (1, 1, 3, 2, 5, 3, 7, 4), as in row 2. - Clark Kimberling, Aug 04 2025
MATHEMATICA
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*)
(* Clark Kimberling, Jul 04 2013 *)
PROG
(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
(SageMath) flatten([[numerator((n-k+1)/k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Jan 12 2022
CROSSREFS
KEYWORD
AUTHOR
Franklin T. Adams-Watters, Sep 11 2005
EXTENSIONS
Keyword tabl added by Franklin T. Adams-Watters, Sep 02 2009
STATUS
approved
