A195278 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.
1, 1, 9, 1, 25, 171, 1, 49, 2125, 6939, 1, 81, 11319, 626525, 609309, 1, 121, 39609, 12608631, 649112125, 116330103, 1, 169, 107811, 119743353, 68604760497, 2375645170875, 48439766655, 1, 225, 248261, 724789395, 2266745635377
Offset: 1
Examples
Some solutions for n=6, k=5 ...0..................0..................0..................0 ...3..0...............0.-4..............-4.-4...............2.-2 ...4..5.10...........-3..1.-4...........-3.-3..0............3..0.-2 ...7..5..7..4........-1.-3.-4.-8.........2..0..1.-3.........7..5..1..1 ...2..2..3..0..5.....-5..0.-5.-6.-9......6..2..3.-2..3......3..6..4..2.-3 ..-2..1..0..1..1.-1..-2..0.-4.-6-10.-5..11..6..1.-4.-1..4...1..4..3..0.-5.-8
Links
- R. H. Hardin, Table of n, a(n) for n = 1..74
Formula
Empirical for rows:
T(1,k) = 1;
T(2,k) = 4*k^2 + 4*k + 1;
T(3,k) = (64/3)*k^5 + (160/3)*k^4 + 56*k^3 + (92/3)*k^2 + (26/3)*k + 1;
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;
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;
Comments