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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

%I #12 Jun 06 2017 11:55:00

%S 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,

%T 30,60,60,42,14,0,9,28,42,105,120,138,78,18,0,10,36,56,168,210,340,

%U 252,144,28,0,11,45,72,252,336,705,620,600,234,39,0

%N 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.

%C Number of primitive (period n) periodic palindromes of length n using a maximum of k different symbols.

%D 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]

%H Andrew Howroyd, <a href="/A284856/b284856.txt">Table of n, a(n) for n = 1..1275</a>

%F T(n, k) = Sum_{d | n} mu(n/d) * A284855(d, k).

%e Table starts:

%e 1 2 3 4 5 6 7 8 9 10 ...

%e 0 1 3 6 10 15 21 28 36 45 ...

%e 0 2 6 12 20 30 42 56 72 90 ...

%e 0 3 12 30 60 105 168 252 360 495 ...

%e 0 6 24 60 120 210 336 504 720 990 ...

%e 0 7 42 138 340 705 1302 2212 3528 5355 ...

%e 0 14 78 252 620 1290 2394 4088 6552 9990 ...

%e 0 18 144 600 1800 4410 9408 18144 32400 54450 ...

%e 0 28 234 1008 3100 7740 16758 32704 58968 99900 ...

%e 0 39 456 2490 9240 26985 66864 146916 294480 548955 ...

%e ...

%t b[d_, k_] := If[EvenQ[d], (k^(d/2) + k^(d/2 + 1))/2, k^((d + 1)/2)];

%t a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k] &];

%t Table[a[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 06 2017, translated from PARI *)

%o (PARI)

%o b(d,k) = if(d % 2 == 0, (k^(d/2) + k^(d/2+1))/2, k^((d+1)/2));

%o a(n,k) = sumdiv(n,d, moebius(n/d) * b(d,k));

%o for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print(););

%Y Columns 2-6 are: A056493, A056494, A056495, A056496, A056497.

%K nonn,tabl

%O 1,2

%A _Andrew Howroyd_, Apr 04 2017

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Last modified March 29 09:42 EDT 2024. Contains 371268 sequences. (Running on oeis4.)