A346793 Triangle T(n,m) read by rows: the number of n X m arrays with nonnegative integers, zeros on the border rows/columns and maximum difference one between any entry and its 4 neighbors.

1, 1, 2, 1, 4, 16, 1, 8, 64, 528, 1, 16, 256, 4368, 75536, 1, 32, 1024, 36176, 1312656, 48185392, 1, 64, 4096, 299664, 22844432, 1776652720, 139414770480, 1, 128, 16384, 2482384, 397748880, 65621158928, 10984586881360, 1852311458443344, 1, 256, 65536, 20563984, 6926263568

Offset

1,3

Comments

The arrays a(i,j) which are counted have constant surrounding borders clamped at a(i,1) = a(i,m) = a(1,j) = a(n,j) =0 , all a(i,j)>=0 and limited slopes |a(i,j)-a(i+1,j)| <= 1, |a(i,j)-a(i-1,j)| <= 1, |a(i,j)-a(i,j+1)| <= 1 and |a(i,j)-a(i,j-1)| <= 1. All rows and columns of the T(n,m) have rational generating functions.

Links

Table of n, a(n) for n=1..41.

R. J. Mathar, Motzkin Islands: a 3-dimensional embeddng of Motzkin paths, vixra:2009.0152 (2020) Table 2.

Formula

T(n,m) = T(m,n).

T(n,m) = 2^((m-1)*(n-1)), 1<=m<=3, n>=1.

T(4,m) = 9*T(4,m-1) -4*T(4,m-2) -16*T(4,m-3).

T(5,m) = 21*T(5,m-1) -52*T(5,m-2) -184*T(5,m-3) +32*T(5,m-4) +128*T(5,m-5).

Example

The triangle starts with n>=1, 1<=m<=n as:
  1
  1    2
  1    4     16
  1    8     64       528
  1   16    256      4368        75536
  1   32   1024     36176      1312656       48185392
  1   64   4096    299664     22844432     1776652720     139414770480
  1  128  16384   2482384    397748880    65621158928   10984586881360 ...
  1  256  65536  20563984   6926263568  2425367471888  867077331528016 ...

Crossrefs

Sequence in context: A032174 A212267 A344110 * A087801 A238454 A025228

Adjacent sequences: A346790 A346791 A346792 * A346794 A346795 A346796

Keyword

nonn,tabl

Author

R. J. Mathar, Aug 04 2021

Status

approved