A195278 - OEIS

%I #8 Jun 18 2022 17:34:15

%S 1,1,9,1,25,171,1,49,2125,6939,1,81,11319,626525,609309,1,121,39609,

%T 12608631,649112125,116330103,1,169,107811,119743353,68604760497,

%U 2375645170875,48439766655,1,225,248261,724789395,2266745635377

%N T(n,k) = Number of lower triangles of an n X n integer array with each element differing from all of its vertical and horizontal neighbors by k or less and triangles differing by a constant counted only once.

%C Table starts

%C .........1.............1................1..................1

%C .........9............25...............49.................81

%C .......171..........2125............11319..............39609

%C ......6939........626525.........12608631..........119743353

%C ....609309.....649112125......68604760497......2266745635377

%C .116330103.2375645170875.1833160598134397.270149651796118149

%H R. H. Hardin, <a href="/A195278/b195278.txt">Table of n, a(n) for n = 1..74</a>

%F Empirical for rows:

%F T(1,k) = 1;

%F T(2,k) = 4*k^2 + 4*k + 1;

%F T(3,k) = (64/3)*k^5 + (160/3)*k^4 + 56*k^3 + (92/3)*k^2 + (26/3)*k + 1;

%F T(4,k) = (49024/315)*k^9 + (24512/35)*k^8 + (457792/315)*k^7 + (81824/45)*k^6 + (67912/45)*k^5 + (38756/45)*k^4 + (11832/35)*k^3 + (27752/315)*k^2 + (1454/105)*k + 1;

%F T(5,k) = (735698944/467775)*k^14 + (735698944/66825)*k^13 + (5736600832/155925)*k^12 + (36310211072/467775)*k^11 + (4906182016/42525)*k^10 + (603389056/4725)*k^9 + (4615314688/42525)*k^8 + (3060288256/42525)*k^7 + (176877304/4725)*k^6 + (647154728/42525)*k^5 + (2229757561/467775)*k^4 + (174277454/155925)*k^3 + (1380143/7425)*k^2 + (68144/3465)*k + 1;

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

%e ...0..................0..................0..................0

%e ...3..0...............0.-4..............-4.-4...............2.-2

%e ...4..5.10...........-3..1.-4...........-3.-3..0............3..0.-2

%e ...7..5..7..4........-1.-3.-4.-8.........2..0..1.-3.........7..5..1..1

%e ...2..2..3..0..5.....-5..0.-5.-6.-9......6..2..3.-2..3......3..6..4..2.-3

%e ..-2..1..0..1..1.-1..-2..0.-4.-6-10.-5..11..6..1.-4.-1..4...1..4..3..0.-5.-8

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_, Sep 14 2011