|
| |
|
|
A054630
|
|
T(n,k) = Sum_{d|k} phi(d)*n^(k/d)/k, triangle read by rows, T(n,k) for n >= 1 and 1 <= k <= n.
|
|
9
|
|
|
|
1, 2, 3, 3, 6, 11, 4, 10, 24, 70, 5, 15, 45, 165, 629, 6, 21, 76, 336, 1560, 7826, 7, 28, 119, 616, 3367, 19684, 117655, 8, 36, 176, 1044, 6560, 43800, 299600, 2097684, 9, 45, 249, 1665, 11817, 88725, 683289, 5381685, 43046889, 10, 55, 340, 2530, 20008, 166870, 1428580, 12501280, 111111340, 1000010044
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
The programs by Peter Luschny below can generate all n-ary necklaces of length k (and all k-ary necklaces of length n) for any positive integer values of n and k, not just for 1 <= k <= n.
From the examples below, we see that the number of 4-ary necklaces of length 3 equals the number of 3-ary necklaces of length 4. The question is whether there are other pairs (n, k) of distinct positive integers such that the number of n-ary necklaces of length k equals the number of k-ary necklaces of length n.
(End)
|
|
|
REFERENCES
|
D. E. Knuth, Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, Addison-Wesley, 2005.
|
|
|
LINKS
|
F. Ruskey, C. Savage, and T. M. Y. Wang, Generating necklaces, Journal of Algorithms, 13(3), 1992, 414-430.
|
|
|
FORMULA
|
T(n,k) = (1/k)*Sum_{i=1..k} n^gcd(i, k). - Peter Luschny, Sep 10 2018
|
|
|
EXAMPLE
|
Triangle starts:
1;
2, 3;
3, 6, 11;
4, 10, 24, 70;
5, 15, 45, 165, 629;
6, 21, 76, 336, 1560, 7826;
The 24 necklaces over {0,1,2} of length 4 are:
0000,0001,0002,0011,0012,0021,0022,0101,0102,0111,0112,0121,
0122,0202,0211,0212,0221,0222,1111,1112,1122,1212,1222,2222.
The 24 necklaces over {0,1,2,3} of length 3 are:
000,001,002,003,011,012,013,021,022,023,031,032,
033,111,112,113,122,123,132,133,222,223,233,333.
|
|
|
MAPLE
|
T := (n, k) -> add(n^igcd(i, k), i=1..k)/k:
|
|
|
MATHEMATICA
|
T[n_, k_] := 1/k Sum[EulerPhi[d] n^(k/d), {d, Divisors[k]}];
|
|
|
PROG
|
(Sage)
def A054630(n, k): return (1/k)*add(euler_phi(d)*n^(k/d) for d in divisors(k))
for n in (1..9):
(Julia)
A054630(n::Int, k::Int) = div(sum(n^gcd(i, k) for i in 1:k), k)
for n in 1:6
println([A054630(n, k) for k in 1:n])
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
