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A284856
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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.
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10
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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
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OFFSET
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1,2
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COMMENTS
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Number of primitive (period n) periodic palindromes of length n using a maximum of k different symbols.
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REFERENCES
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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]
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LINKS
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FORMULA
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T(n, k) = Sum_{d | n} mu(n/d) * A284855(d, k).
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EXAMPLE
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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 ...
...
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MATHEMATICA
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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 *)
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PROG
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(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(); );
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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