A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.
1, 2, 0, 3, 1, 0, 4, 3, 4, 0, 5, 6, 15, 7, 0, 6, 10, 36, 39, 18, 0, 7, 15, 70, 126, 132, 29, 0, 8, 21, 120, 310, 540, 357, 70, 0, 9, 28, 189, 645, 1620, 2034, 1131, 126, 0, 10, 36, 280, 1197, 3990, 7790, 8316, 3276, 266, 0
Offset: 1
Examples
Table starts: 1 2 3 4 5 6 7 8 ... 0 1 3 6 10 15 21 28 ... 0 4 15 36 70 120 189 280 ... 0 7 39 126 310 645 1197 2044 ... 0 18 132 540 1620 3990 8568 16632 ... 0 29 357 2034 7790 23295 58779 131012 ... 0 70 1131 8316 39370 140610 412965 1050616 ... 0 126 3276 32760 195300 839790 2882376 8388576 ... ...
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
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
Programs
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Mathematica
b[n_, k_] := (k^n + k^Ceiling[n/2])/2; a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k]&]; Table[a[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017, translated from PARI *) -
PARI
b(n,k) = (k^n + k^(ceil(n/2))) / 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(););
Formula
T(n, k) = Sum_{d | n} mu(n/d) * (k^n + k^(ceiling(n/2))) / 2.
Comments