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A284856
Array read by antidiagonals: T(n,k) = number of aperiodic necklaces (Lyndon words) with n beads and k colors that are the same when turned over.
10
1, 2, 0, 3, 1, 0, 4, 3, 2, 0, 5, 6, 6, 3, 0, 6, 10, 12, 12, 6, 0, 7, 15, 20, 30, 24, 7, 0, 8, 21, 30, 60, 60, 42, 14, 0, 9, 28, 42, 105, 120, 138, 78, 18, 0, 10, 36, 56, 168, 210, 340, 252, 144, 28, 0, 11, 45, 72, 252, 336, 705, 620, 600, 234, 39, 0
OFFSET
1,2
COMMENTS
Number of primitive (period n) periodic palindromes of length n using a maximum of k different symbols.
REFERENCES
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
LINKS
FORMULA
T(n, k) = Sum_{d | n} mu(n/d) * A284855(d, k).
EXAMPLE
Table starts:
1 2 3 4 5 6 7 8 9 10 ...
0 1 3 6 10 15 21 28 36 45 ...
0 2 6 12 20 30 42 56 72 90 ...
0 3 12 30 60 105 168 252 360 495 ...
0 6 24 60 120 210 336 504 720 990 ...
0 7 42 138 340 705 1302 2212 3528 5355 ...
0 14 78 252 620 1290 2394 4088 6552 9990 ...
0 18 144 600 1800 4410 9408 18144 32400 54450 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 39 456 2490 9240 26985 66864 146916 294480 548955 ...
...
MATHEMATICA
b[d_, k_] := If[EvenQ[d], (k^(d/2) + k^(d/2 + 1))/2, k^((d + 1)/2)];
a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k] &];
Table[a[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 06 2017, translated from PARI *)
PROG
(PARI)
b(d, k) = if(d % 2 == 0, (k^(d/2) + k^(d/2+1))/2, k^((d+1)/2));
a(n, k) = sumdiv(n, d, moebius(n/d) * b(d, k));
for(n=1, 10, for(k=1, 10, print1( a(n, k), ", "); ); print(); );
CROSSREFS
Columns 2-6 are: A056493, A056494, A056495, A056496, A056497.
Sequence in context: A321980 A208544 A208535 * A276550 A294438 A074650
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 04 2017
STATUS
approved