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A284823
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Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).
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10
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1, 2, 0, 3, 0, 0, 4, 0, 2, 0, 5, 0, 6, 2, 0, 6, 0, 12, 6, 6, 0, 7, 0, 20, 12, 24, 4, 0, 8, 0, 30, 20, 60, 18, 14, 0, 9, 0, 42, 30, 120, 48, 78, 12, 0, 10, 0, 56, 42, 210, 100, 252, 72, 28, 0, 11, 0, 72, 56, 336, 180, 620, 240, 234, 24, 0, 12, 0, 90, 72, 504, 294, 1290, 600, 1008, 216, 62
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OFFSET
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1,2
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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) * k^(ceiling(d/2)).
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EXAMPLE
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Table starts:
1 2 3 4 5 6 7 8 9 10 ...
0 0 0 0 0 0 0 0 0 0 ...
0 2 6 12 20 30 42 56 72 90 ...
0 2 6 12 20 30 42 56 72 90 ...
0 6 24 60 120 210 336 504 720 990 ...
0 4 18 48 100 180 294 448 648 900 ...
0 14 78 252 620 1290 2394 4088 6552 9990 ...
0 12 72 240 600 1260 2352 4032 6480 9900 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 24 216 960 3000 7560 16464 32256 58320 99000 ...
...
Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).
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MATHEMATICA
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T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017 *)
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PROG
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(PARI)
a(n, k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
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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