A195220 - OEIS

%I #12 Oct 07 2019 03:05:18

%S 1,1,7,1,19,91,1,37,1047,2277,1,61,5453,176471,111031,1,91,18903,

%T 3395245,92031109,10654607,1,127,51205,31640829,9032683465,

%U 149824887097,2021888119,1,169,117585,189677411,289301569283,103565705397639

%N 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.

%C Table starts

%C         1            1               1                 1                  1

%C         7           19              37                61                 91

%C        91         1047            5453             18903              51205

%C      2277       176471         3395245          31640829          189677411

%C    111031     92031109      9032683465      289301569283      4677360495205

%C  10654607 149824887097 103565705397639 14572563308953245 774355028021195459

%C 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

%H R. H. Hardin, <a href="/A195220/b195220.txt">Table of n, a(n) for n = 1..86</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrhart_polynomial">Ehrhart polynomial</a>

%F Empirical for rows:

%F T(1,k) = 1

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

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

%F 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

%F 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

%e Some solutions for n=6, k=5:

%e    0                  0                  0                  0

%e    4  4               2  2               2  1               4  5

%e    6  7  7            7  6  5           -3 -2  1            5  8  7

%e   10  8 12  7         4  7  6  1        -6 -1 -4 -2         8  9  5  7

%e   10 12 11 12  9      2  3  5  1  0     -1 -1 -1 -1 -2      5  5  8  9  8

%e    7  7  8 12  9  5   1  3  5  2  4  5  -6 -3  0  0  1 -3   0  3  8  8 10  6

%Y Row 2 is A003215.

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_, Sep 13 2011