A195220 T(n,k) is the number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by k or less and triangles differing by a constant counted only once.

1, 1, 7, 1, 19, 91, 1, 37, 1047, 2277, 1, 61, 5453, 176471, 111031, 1, 91, 18903, 3395245, 92031109, 10654607, 1, 127, 51205, 31640829, 9032683465, 149824887097, 2021888119, 1, 169, 117585, 189677411, 289301569283, 103565705397639

Offset: 1

Author

R. H. Hardin, Sep 13 2011

Keywords

• nonn
• tabl

Comments

Table starts

1 1 1 1 1

7 19 37 61 91

91 1047 5453 18903 51205

2277 176471 3395245 31640829 189677411

111031 92031109 9032683465 289301569283 4677360495205

10654607 149824887097 103565705397639 14572563308953245 774355028021195459

T(n,k) is the number of integer lattice points in kP where P is a (n*(n+1)/2-1)-dimensional polytope with vertices whose coordinates are all in {-1,0,1}. Therefore it is an Ehrhart polynomial in k, with degree n*(n+1)/2-1 and rational coefficients. - Robert Israel, Oct 06 2019

Examples

			Some solutions for n=6, k=5:
   0                  0                  0                  0
   4  4               2  2               2  1               4  5
   6  7  7            7  6  5           -3 -2  1            5  8  7
  10  8 12  7         4  7  6  1        -6 -1 -4 -2         8  9  5  7
  10 12 11 12  9      2  3  5  1  0     -1 -1 -1 -1 -2      5  5  8  9  8
   7  7  8 12  9  5   1  3  5  2  4  5  -6 -3  0  0  1 -3   0  3  8  8 10  6
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..86
• Wikipedia, Ehrhart polynomial

Crossrefs

Row 2 is A003215.

Formula

Empirical for rows:

T(1,k) = 1

T(2,k) = 3*k^2 + 3*k + 1

T(3,k) = (301/30)*k^5 + (301/12)*k^4 + (88/3)*k^3 + (227/12)*k^2 + (199/30)*k + 1

T(4,k) = (1207573/30240)*k^9 + (1207573/6720)*k^8 + (1000157/2520)*k^7 + (264247/480)*k^6 + (754417/1440)*k^5 + (338651/960)*k^4 + (2533393/15120)*k^3 + (90763/1680)*k^2 + (901/84)*k + 1

T(5,k) = (3508493543/18345600)*k^14 + (3508493543/2620800)*k^13 + (1116775769537/239500800)*k^12 + (422094048023/39916800)*k^11 + (377328209183/21772800)*k^10 + (78475421219/3628800)*k^9 + (1073748492569/50803200)*k^8 + (19848770813/1209600)*k^7 + (221251862417/21772800)*k^6 + (18121075223/3628800)*k^5 + (10435002133/5443200)*k^4 + (505904317/907200)*k^3 + (8793472607/75675600)*k^2 + (1397863/90090)*k + 1