# [seqfan] Isosceles Colorings

Ron Hardin rhhardin at att.net
Tue May 17 14:48:51 CEST 2016

```Colorings of an nXk array with no 3 vertices of an isosceles (and isosceles right) triangle being the same color.
You get a "finite" T(n,k) of course, except for leading rows and columns.

/tmp/fok
T(n,k)=Number of nXk 0..1 arrays with no three equal values forming an isosceles triangle, and new values introduced in 0..1 order
Table starts
...1..2.4.8.16.32.64.128.256.512.1024.2048.4096.8192
...2..3.4.6..5..7..7...9...9..11...11...13...13.....
...4..4.0.0..0..0..0...0...0...0....0....0..........
...8..6.0.0..0..0..0...0...0...0....0...............
..16..5.0.0..0..0..0...0...0...0....................
..32..7.0.0..0..0..0...0...0........................
..64..7.0.0..0..0..0...0............................
.128..9.0.0..0..0..0................................
.256..9.0.0..0..0...................................
.512.11.0.0..0......................................

/tmp/foi
T(n,k)=Number of nXk 0..2 arrays with no three equal values forming an isosceles triangle, and new values introduced in 0..2 order
Table starts
....1......2...5..14...41..122...365..1094...3281...9842...29525...88574
....2......9..50.285.1129.4791.17219.63640.182143.598332.1622713.4807733
....5.....50.136.111...97..235...197...389....305....589.....447.....823
...14....285.111...0....0....0.....0.....0......0......0.......0........
...41...1129..97...0....0....0.....0.....0......0......0................
..122...4791.235...0....0....0.....0.....0......0.......................
..365..17219.197...0....0....0.....0.....0..............................
.1094..63640.389...0....0....0.....0....................................
.3281.182143.305...0....0....0..........................................
.9842.598332.589...0....0...............................................

/tmp/foj
T(n,k)=Number of nXk 0..3 arrays with no three equal values forming an isosceles triangle, and new values introduced in 0..3 order
Table starts
.....1.........2..........5....15.....51.....187......715......2795.....11051
.....2........10........105..1264..13045..136063..1273961..11936399..95697853
.....5.......105.......1946.24587.211121.2116450.15084593.112714143.511843593
....15......1264......24587.49534..36817...40569....36570.....90627.....89429
....51.....13045.....211121.36817....356......10........0.........0..........
...187....136063....2116450.40569.....10.......0........0....................
...715...1273961...15084593.36570......0.......0.............................
..2795..11936399..112714143.90627......0.....................................
.11051..95697853..511843593.89429............................................
.43947.800561388.2841058530..................................................
Empirical for column k:
k=1: a(n)=7*a(n-1)-14*a(n-2)+8*a(n-3)

/tmp/fol
T(n,k)=Number of nXk 0..1 arrays with no three equal values forming an isosceles right triangle, and new values introduced in 0..1 order
Table starts
...1..2.4.8.16.32.64.128.256.512.1024.2048.4096.8192
...2..3.4.6..9.14.22..35..56..90..145..234..378.....
...4..4.0.0..0..0..0...0...0...0....0....0..........
...8..6.0.0..0..0..0...0...0...0....0...............
..16..9.0.0..0..0..0...0...0...0....................
..32.14.0.0..0..0..0...0...0........................
..64.22.0.0..0..0..0...0............................
.128.35.0.0..0..0..0................................
.256.56.0.0..0..0...................................
.512.90.0.0..0......................................
Empirical for column k:
k=1: a(n)=2*a(n-1)
k=2: a(n)=2*a(n-1)-a(n-3)

/tmp/fom
T(n,k)=Number of nXk 0..2 arrays with no three equal values forming an isosceles right triangle, and new values introduced in 0..2 order
Table starts
....1.......2.....5..14...41..122...365...1094....3281....9842.29525
....2.......9....50.285.1617.9188.52193.296511.1684466.9569425......
....5......50...264.435..546.1209..3272...8412...20634..............
...14.....285...435...8....1....0.....0......0......................
...41....1617...546...1....0....0.....0.............................
..122....9188..1209...0....0....0...................................
..365...52193..3272...0....0........................................
.1094..296511..8412...0.............................................
.3281.1684466.20634.................................................
.9842.9569425.......................................................
Empirical for column k:
k=2: a(n)=6*a(n-1)-11*a(n-3)+4*a(n-4) for n>5
k=3: [order 40] for n>49

/tmp/fon
T(n,k)=Number of nXk 0..3 arrays with no three equal values forming an isosceles right triangle, and new values introduced in 0..3 order
Table starts
.....1..........2.............5.........15......51......187.......715
.....2.........10...........105.......1264...15418...188465...2304141
.....5........105..........2766......49702..784878.13322548.232027264
....15.......1264.........49702.....543906.4010258.16365681..55643290
....51......15418........784878....4010258.5029904...725420.....30060
...187.....188465......13322548...16365681..725420.....2890........12
...715....2304141.....232027264...55643290...30060.......12.........0
..2795...28171070....4056338728..210405148....1337........2.........0
.11051..344428103...70402715640..904155053......54........0.........0
.43947.4211086174.1221782263634.4384022592......11........0..........
Empirical for column k:
k=1: a(n)=7*a(n-1)-14*a(n-2)+8*a(n-3)
k=2: [order 7] for n>8

rhhardin at mindspring.com rhhardin at att.net (either)
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