A215228 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").

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, 0, 9, 56, 210, 600, 720, 408, 0, 0, 10, 72, 336, 1260, 2520, 2940, 840, 24, 0, 11, 90, 504, 2352, 6720, 12600, 10080, 2448, 0, 0, 12, 110, 720, 4032, 15120, 40110, 57960, 38640

Offset

1,1

Comments

Table starts

2 3 4 5 6 7 8 9 10

2 6 12 20 30 42 56 72 90

0 6 24 60 120 210 336 504 720

0 12 72 240 600 1260 2352 4032 6480

0 0 120 720 2520 6720 15120 30240 55440

0 18 408 2940 12600 40110 105168 240408 496080

0 0 840 10080 57960 228480 710640 1874880 4379760

0 24 2448 38640 280560 1338120 4883424 14783328 38962080

0 0 5760 140400 1330560 7761600 33384960 116212320 345945600

0 0 15960 529440 6394680 45291120 228945360 915183360 3075040080

0 66 39864 1956900 30548760 263674950 1568401296 7203324744

0 72 108024 7335840 146516040 1537291560 10751253072

Empirical: row n is a polynomial of degree n.

Coefficients for rows 1-10, highest power first:

1 1

1 1 0

1 0 -1 0

1 0 -1 0 0

1 0 -5 0 4 0

1 0 -6 5 5 -5 0

1 0 -7 0 14 0 -8 0

1 0 -8 0 27 -12 -20 12 0

1 0 -9 0 27 0 -31 0 12 0

1 0 -10 0 35 9 -60 -25 34 16 0

Row n is divisible by n.

Column k is divisible by k+1.

From Robert Israel, Nov 23 2017: (Start)

Row n is a monic polynomial of degree n.

Proof: Let b(j,n,k) be the number of such arrays taking exactly j different values.

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.

In particular, b(n,n,n-1) = n!, so b(n,n,k) has degree n and leading coefficient 1. (End)

Links

R. H. Hardin, Table of n, a(n) for n = 1..165

Example

Some solutions for n=5, k=4:
  3  0  1  1  1  0  4  4  0  1  3  2  2  3  1  0
  2  4  0  3  0  4  3  2  2  2  4  0  4  4  4  1
  0  2  2  2  2  3  0  3  1  4  0  4  3  1  0  0
  3  0  3  0  3  1  3  4  4  0  3  0  0  3  4  2
  1  3  2  4  0  2  1  0  1  4  2  1  4  0  2  3

Crossrefs

Column 2 is A066297.

Row 2 is A002378.

Row 3 is A007531(n+1).

Row 4 is A047928(n+1).

Row 5 is A052787(n+2).

Sequence in context: A308158 A325349 A320054 * A182309 A043263 A321326

Adjacent sequences: A215225 A215226 A215227 * A215229 A215230 A215231

Keyword

nonn,tabl

Author

R. H. Hardin, Aug 06 2012

Status

approved