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%I #16 Jan 22 2022 23:41:46
%S 2,3,2,4,6,0,5,12,6,0,6,20,24,12,0,7,30,60,72,0,0,8,42,120,240,120,18,
%T 0,9,56,210,600,720,408,0,0,10,72,336,1260,2520,2940,840,24,0,11,90,
%U 504,2352,6720,12600,10080,2448,0,0,12,110,720,4032,15120,40110,57960,38640
%N T(n,k) = number of length-n 0..k arrays connected end-around, with no sequence of L<n elements immediately followed by itself (periodic "squarefree").
%C Table starts
%C 2 3 4 5 6 7 8 9 10
%C 2 6 12 20 30 42 56 72 90
%C 0 6 24 60 120 210 336 504 720
%C 0 12 72 240 600 1260 2352 4032 6480
%C 0 0 120 720 2520 6720 15120 30240 55440
%C 0 18 408 2940 12600 40110 105168 240408 496080
%C 0 0 840 10080 57960 228480 710640 1874880 4379760
%C 0 24 2448 38640 280560 1338120 4883424 14783328 38962080
%C 0 0 5760 140400 1330560 7761600 33384960 116212320 345945600
%C 0 0 15960 529440 6394680 45291120 228945360 915183360 3075040080
%C 0 66 39864 1956900 30548760 263674950 1568401296 7203324744
%C 0 72 108024 7335840 146516040 1537291560 10751253072
%C Empirical: row n is a polynomial of degree n.
%C Coefficients for rows 1-10, highest power first:
%C 1 1
%C 1 1 0
%C 1 0 -1 0
%C 1 0 -1 0 0
%C 1 0 -5 0 4 0
%C 1 0 -6 5 5 -5 0
%C 1 0 -7 0 14 0 -8 0
%C 1 0 -8 0 27 -12 -20 12 0
%C 1 0 -9 0 27 0 -31 0 12 0
%C 1 0 -10 0 35 9 -60 -25 34 16 0
%C Row n is divisible by n.
%C Column k is divisible by k+1.
%C From _Robert Israel_, Nov 23 2017: (Start)
%C Row n is a monic polynomial of degree n.
%C Proof: Let b(j,n,k) be the number of such arrays taking exactly j different values.
%C Then T(n,k) = Sum_{j <= n} b(j,n,k). But since the j values may be any combination of 0..k taken j at a time, b(j,n,k) = binomial(k+1,j)* b(j,n,j-1) which (if nonzero) is a polynomial in k of degree j.
%C In particular, b(n,n,n-1) = n!, so b(n,n,k) has degree n and leading coefficient 1. (End)
%H R. H. Hardin, <a href="/A215228/b215228.txt">Table of n, a(n) for n = 1..165</a>
%e Some solutions for n=5, k=4:
%e 3 0 1 1 1 0 4 4 0 1 3 2 2 3 1 0
%e 2 4 0 3 0 4 3 2 2 2 4 0 4 4 4 1
%e 0 2 2 2 2 3 0 3 1 4 0 4 3 1 0 0
%e 3 0 3 0 3 1 3 4 4 0 3 0 0 3 4 2
%e 1 3 2 4 0 2 1 0 1 4 2 1 4 0 2 3
%Y Column 2 is A066297.
%Y Row 2 is A002378.
%Y Row 3 is A007531(n+1).
%Y Row 4 is A047928(n+1).
%Y Row 5 is A052787(n+2).
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Aug 06 2012