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A226691
Irregular table read by rows: T(m,n) = Denominators of minimal distance between m elements of the Farey sequence F_n.
2
1, 1, 1, 2, 1, 1, 6, 3, 3, 1, 1, 12, 4, 12, 2, 4, 1, 1, 20, 15, 5, 10, 5, 15, 20, 5, 5, 1, 1, 30, 12, 6, 30, 3, 30, 2, 12, 30, 3, 6, 1, 1, 42, 35, 28, 7, 21, 35, 7, 14, 7, 35, 21, 7, 28, 35, 42, 7, 7, 1, 1, 56, 24, 40, 8, 56, 24, 4, 40, 56, 8, 56, 40, 2, 24, 56, 8, 40, 24, 56
OFFSET
1,4
COMMENTS
The first element of each row is 0 (because the distance from any element to itself is 0), the last element of each row is 1 (because the distance from 0 to 1 is 1). These give denominators of 1 and 1.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
Table begins:
0, 1;
0, 1/2, 1;
0, 1/6, 1/3, 2/3, 1;
0, 1/12, 1/4, 5/12, 1/2, 3/4, 1;
Consider the third row. The Farey series for 3 is (0, 1/3, 1/2, 2/3, 1). 0 is always first. The closest two elements are 1/3 and 1/2, so 1/6 is next. The closest three are 1/3 and 2/3 with distance 1/3. Including either endpoint gives distance 2/3; both gives distance 1.
PROG
(PARI) Farey(n)=my(v=List([0, 1])); for(a=2, n, for(b=1, a-1, listput(v, b/a))); vecsort(Vec(v), , 8)
f(n)=my(F=Farey(n)); vector(#F, k, my(b=1); for(i=k, #F, b=min(F[i]-F[i-k+1], b)); b)
for(n=1, 9, v=apply(denominator, f(n)); for(i=1, #v, print1(v[i]", ")))
CROSSREFS
Cf. A226690.
Sequence in context: A284308 A369435 A172400 * A158389 A186287 A318393
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved