A167682 -id:A167682 - cp's OEIS frontend

A229534 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally, diagonally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

0, 1, 0, 2, 4, 0, 6, 8, 20, 0, 16, 36, 58, 84, 0, 40, 112, 361, 356, 324, 0, 96, 368, 1588, 3064, 2038, 1188, 0, 224, 1152, 7460, 19276, 24344, 11184, 4212, 0, 512, 3568, 33136, 130854, 221096, 185808, 59626, 14580, 0, 1152, 10880, 146300, 833108, 2171944

Offset: 1

R. H. Hardin, Sep 25 2013

Table starts
.0.....1......2........6........16.........40...........96...........224
.0.....4......8.......36.......112........368.........1152..........3568
.0....20.....58......361......1588.......7460........33136........146300
.0....84....356.....3064.....19276.....130854.......833108.......5305746
.0...324...2038....24344....221096....2171944.....19965136.....184319130
.0..1188..11184...185808...2451728...34811238....463976296....6218438820
.0..4212..59626..1379512..26566266..544403948..10551803060..205336122417
.0.14580.311260.10036352.283010776.8359264560.236116939092.6668992563052

			Some solutions for n=3, k=4:
  0 1 0 1     0 1 0 2     0 1 0 0     0 1 0 0     0 1 2 0
  0 2 0 2     1 2 0 2     0 1 2 1     2 1 2 1     0 1 2 0
  2 1 0 1     1 2 0 2     2 1 2 1     0 1 2 1     2 1 0 1

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 2 is A167682(n-1).

Row 1 is A057711(n-1).

Empirical for column k:
k=1: a(n) = a(n-1).
k=2: a(n) = 6*a(n-1) - 9*a(n-2) for n > 3.
k=3: a(n) = 10*a(n-1) - 29*a(n-2) + 20*a(n-3) - 4*a(n-4) for n > 5.
k=4: a(n) = 14*a(n-1) - 57*a(n-2) + 56*a(n-3) - 16*a(n-4) for n > 5.
k=5: [order 12] for n > 13.
k=6: [order 18] for n > 19.
k=7: [order 38] for n > 39.
Empirical for row n:
n=1: a(n) = 4*a(n-1) - 4*a(n-2) for n > 4.
n=2: a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4).
n=3: a(n) = 6*a(n-1) - a(n-2) - 28*a(n-3) - 4*a(n-4) + 16*a(n-5) - 4*a(n-6) for n > 8.
n=4: [order 12] for n > 14.
n=5: [order 20] for n > 22.
n=6: [order 46] for n > 48.
n=7: [order 92] for n > 94.

A229460 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and vertically with exactly one mistake and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 6, 20, 20, 6, 16, 84, 140, 84, 16, 40, 324, 863, 863, 324, 40, 96, 1188, 4962, 7940, 4962, 1188, 96, 224, 4212, 27313, 68790, 68790, 27313, 4212, 224, 512, 14580, 145932, 573342, 903332, 573342, 145932, 14580, 512, 1152, 49572, 763031

Offset: 1

R. H. Hardin, Sep 24 2013

Table starts
...0.....1......2........6.........16..........40............96.............224
...1.....4.....20.......84........324........1188..........4212...........14580
...2....20....140......863.......4962.......27313........145932..........763031
...6....84....863.....7940......68790......573342.......4651079........36985536
..16...324...4962....68790.....903332....11451686.....141595454......1718447506
..40..1188..27313...573342...11451686...221410052....4182294415.....77626332302
..96..4212.145932..4651079..141595454..4182294415..120864516084...3435347473308
.224.14580.763031.36985536.1718447506.77626332302.3435347473308.149656305350148

			Some solutions for n=3, k=4:
  0 1 0 2     0 1 0 2     0 1 0 2     0 1 0 2     0 1 0 2
  2 1 2 1     2 1 2 0     2 2 1 0     1 0 2 1     1 0 2 0
  0 2 1 2     1 2 1 2     0 1 2 1     2 0 1 0     1 2 0 2

R. H. Hardin, Table of n, a(n) for n = 1..420

Column 1 is A057711(n-1).

Column 2 is A167682(n-1).

Empirical for column k:
k=1: a(n) = 4*a(n-1) - 4*a(n-2) for n > 4.
k=2: a(n) = 6*a(n-1) - 9*a(n-2) for n > 3.
k=3: a(n) = 10*a(n-1) - 29*a(n-2) + 20*a(n-3) - 4*a(n-4).
k=4: [order 6] for n > 7.
k=5: [order 10].
k=6: [order 14] for n > 15.
k=7: [order 26].
k=8: [order 38] for n > 39.

A167666 Triangle read by rows given by [1,1,-4,2,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 0, 4, 5, 1, 0, 0, 6, 7, 1, 0, 0, 0, 8, 9, 1, 0, 0, 0, 0, 10, 11, 1, 0, 0, 0, 0, 0, 12, 13, 1, 0, 0, 0, 0, 0, 0, 14, 15, 1, 0, 0, 0, 0, 0, 0, 0, 16, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, 19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 23, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Nov 08 2009

Row sums = A111284(n+1), Diagonal sums = A109613(n).

			Triangle begins :
1 ;
1, 1 ;
2, 3, 1 ;
0, 4, 5, 1 ;
0, 0, 6, 7, 1 ;
0, 0, 0, 8, 9, 1 ;
0, 0, 0, 0, 10, 11, 1 ; ...

Cf. A000012, A005408, A005843

T(n,k) = 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(3,0) = 0, T(3,1) = 4. - Philippe Deléham, Feb 18 2012
G.f.: (1+(1-y)*x+(2+y)*x^2)/(1-y*x)^2. - Philippe Deléham, Feb 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A130779(n), A111284(n+1), A167667(n), A167682(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 18 2012

cp's OEIS Frontend

A229534 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally, diagonally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229460 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and vertically with exactly one mistake and colors introduced in row-major 0..2 order.

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Formula

A167666 Triangle read by rows given by [1,1,-4,2,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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Comments

Examples

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Formula