OFFSET
1,1
COMMENTS
So, T(n, k) is also the index of fraction 1/k in the Farey fractions of order n. - Michel Marcus, Jun 27 2014
Start with array [1,k], and for each integer i from k+1 to n, insert i between every consecutive pair that sums to i. The length of the resulting array is T(n,k). For example, with n=5 and k=2 we have [1,2] -> [1,3,2] -> [1,4,3,2] -> [1,5,4,3,5,2] which has length 6, so T(5,2)=6. This is from a discovery of Leo Moser as described by Martin Gardner. - Curtis Bechtel, Oct 05 2024
REFERENCES
Martin Gardner, The Last Recreations, 1997, chapter 12.
LINKS
Curtis Bechtel, Table of n, a(n) for n = 1..5050
EXAMPLE
Rows: {2}; {3,2}; {5,3,2}; ...; e.g. in row 3, 5 reduced fractions (0/1,1/3,1/2,2/3,1/1) are <=1; 3 are <=1/2; 2 are <=1/3.
Triangle starts:
2;
3, 2;
5, 3, 2;
7, 4, 3, 2;
11, 6, 4, 3, 2;
13, 7, 5, 4, 3, 2;
...
MATHEMATICA
T[n_, k_] := Count[FareySequence[n], f_ /; f <= 1/k];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 25 2018 *)
PROG
(PARI) row(nn) = my(frow = farey(n)); for (k=1, n, print1(vecsearch(frow, 1/k), ", "); ); \\ Michel Marcus, Jun 27 2014
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved