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A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols. 10
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

FORMULA

T(n,k) = Sum_{d | n} mu(n/d) * k^(ceiling(d/2)).

EXAMPLE

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).

MATHEMATICA

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 *)

PROG

(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(); )

CROSSREFS

Columns 2-6 are A056458, A056459, A056460, A056461, A056462.

Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.

Cf. A284826, A284841.

Sequence in context: A245964 A141700 A035205 * A131104 A141701 A167990

Adjacent sequences:  A284820 A284821 A284822 * A284824 A284825 A284826

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 03 2017

STATUS

approved

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Last modified January 18 13:30 EST 2020. Contains 331007 sequences. (Running on oeis4.)