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A339942 Triangle read by rows: T(n,k) is the number of permutations of the cyclic group Z/nZ whose longest embedded arithmetic progression has length k. 1
1, 0, 2, 0, 0, 6, 0, 8, 8, 8, 0, 0, 40, 60, 20, 0, 0, 468, 192, 48, 12, 0, 0, 462, 3150, 1176, 210, 42, 0, 128, 4192, 27872, 6592, 1312, 192, 32, 0, 0, 57402, 182790, 99630, 19656, 2970, 378, 54, 0, 0, 67440, 1795320, 1594640, 146200, 22000, 2840, 320, 40, 0, 0, 61050, 17433130, 17373620, 4289340, 662860, 85910, 9790, 990, 110 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For the case k=2, it can be proved that if n is a power of 2, then T(n,2)=2^{n-1}; otherwise T(n,2)=0 (Lemma 8 of Goh and Zhao (2020)). It can also be shown that T(n,n) = n*phi(n), where phi is the Euler totient function.

LINKS

Table of n, a(n) for n=1..66.

M. K. Goh and R. Y. Zhao, Arithmetic subsequences in a random ordering of an additive set, arXiv:2012.12339 [math.CO], 2020.

FORMULA

T(n,n) = n*A000010(n).

EXAMPLE

Triangle T(n,k) begins:

  n/k 1   2        3         4         5        6      7     8    9  10  11

   1  1

   2  0   2

   3  0   0        6

   4  0   8        8         8

   5  0   0       40        60        20

   6  0   0      468       192        48       12

   7  0   0      462      3150      1176      210     42

   8  0 128     4192     27872      6592     1312    192    32

   9  0   0    57402    182790     99630    19656   2970   378   54

  10  0   0    67440   1795320   1594640   146200  22000  2840  320  40

  11  0   0    61050  17433130  17373620  4289340 662860 85910 9790 990 110

CROSSREFS

Cf. A000010, A002618, A339941.

Sequence in context: A028625 A344441 A221728 * A345366 A278720 A221524

Adjacent sequences:  A339939 A339940 A339941 * A339943 A339944 A339945

KEYWORD

nonn,tabl

AUTHOR

Marcel K. Goh, Dec 23 2020

STATUS

approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)