A052482 -id:A052482 - cp's OEIS frontend

A229637 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally, diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

0, 0, 0, 1, 6, 0, 3, 40, 39, 0, 12, 122, 244, 202, 0, 40, 488, 1109, 1496, 925, 0, 120, 1608, 6031, 10227, 8800, 3924, 0, 336, 5392, 28448, 77620, 89331, 50084, 15795, 0, 896, 17368, 136778, 535671, 960325, 747299, 277996, 61182, 0, 2304, 55232, 633328

Offset: 1

R. H. Hardin, Sep 27 2013

Table starts
.0.....0.......1........3.........12..........40...........120............336
.0.....6......40......122........488........1608..........5392..........17368
.0....39.....244.....1109.......6031.......28448........136778.........633328
.0...202....1496....10227......77620......535671.......3723370.......25022190
.0...925....8800....89331.....960325.....9722206......98015235......960209886
.0..3924...50084...747299...11485716...170405645....2495874984....35693194243
.0.15795..277996..6049298..133784624..2902520386...61836040854..1290897457785
.0.61182.1513104.47723226.1525870912.48303362606.1498317588826.45634751291449

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

R. H. Hardin, Table of n, a(n) for n = 1..287

Column 2 is A229600.

Row 1 is A052482(n-2).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) for n > 5
k=3: a(n) = 15*a(n-1) - 81*a(n-2) + 185*a(n-3) - 162*a(n-4) + 60*a(n-5) - 8*a(n-6) for n > 7.
k=4: [order 6] for n > 9.
k=5: [order 18] for n > 20.
k=6: [order 27] for n > 30.
k=7: [order 57] for n > 60.
Empirical for row n:
n=1: a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 6.
n=2: a(n) = 6*a(n-1) - 6*a(n-2) - 16*a(n-3) + 12*a(n-4) + 24*a(n-5) + 8*a(n-6).
n=3: [order 9] for n > 12.
n=4: [order 18] for n > 21.
n=5: [order 30] for n > 33.
n=6: [order 69] for n > 72.

A229694 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 0, 0, 1, 3, 0, 3, 43, 40, 0, 12, 245, 626, 336, 0, 40, 1171, 5077, 6732, 2304, 0, 120, 5077, 35825, 80757, 62856, 14080, 0, 336, 20691, 230383, 848937, 1125333, 539568, 79872, 0, 896, 80757, 1400413, 8186713, 17724789, 14461173, 4377888, 430080, 0

Offset: 1

R. H. Hardin, Sep 27 2013

			Some solutions for n=3, k=4:
  0 1 2 1     0 1 0 2     0 1 0 2     0 0 1 2     0 1 0 2
  0 1 0 2     0 2 0 2     0 2 1 0     1 0 0 1     0 2 1 1
  1 2 1 1     2 1 2 0     2 2 1 2     2 1 2 0     1 0 2 2
Table starts
.0......0........1..........3...........12............40.............120
.0......3.......43........245.........1171..........5077...........20691
.0.....40......626.......5077........35825........230383.........1400413
.0....336.....6732......80757.......848937.......8186713........75035643
.0...2304....62856....1125333.....17724789.....258006388......3583403667
.0..14080...539568...14461173....342532665....7551515197....159377253183
.0..79872..4377888..175867605...6279934941..210095323918...6749642728251
.0.430080.34105536.2054728053.110801828529.5632122625852.275739382892979

R. H. Hardin, Table of n, a(n) for n = 1..286

Column 2 is A002700(n+1).

Row 1 is A052482(n-2).

Empirical for column k:
k=1: a(n) = a(n-1).
k=2: a(n) = 12*a(n-1) - 48*a(n-2) + 64*a(n-3).
k=3: a(n) = 18*a(n-1) - 108*a(n-2) + 216*a(n-3) for n > 4.
k=4: a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3) for n > 4.
k=5: [order 6] for n > 7.
k=6: [order 9] for n > 11.
k=7: [order 12] for n > 14.
Empirical for row n:
n=1: a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 6.
n=2: a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) for n > 6.
n=3: a(n) = 15*a(n-1) - 81*a(n-2) + 185*a(n-3) - 162*a(n-4) + 60*a(n-5) - 8*a(n-6) for n > 10.
n=4: [order 9] for n > 17.
n=5: [order 21] for n > 27.
n=6: [order 29] for n > 39.
n=7: [order 86] for n > 94.

A080929 Sequence associated with a(n) = 2a(n-1) + k(k+2)*a(n-2).

Original entry on oeis.org

1, 3, 12, 40, 120, 336, 896, 2304, 5760, 14080, 33792, 79872, 186368, 430080, 983040, 2228224, 5013504, 11206656, 24903680, 55050240, 121110528, 265289728, 578813952, 1258291200, 2726297600, 5888802816, 12683575296, 27246198784

Offset: 0

Paul Barry, Feb 26 2003

The third column of number triangle A080928.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Essentially the same as A052482.

Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.

Concatenation([1], List([1..30], n-> 2^(n-1)*Binomial(n+2,2))); # G. C. Greubel, Jul 23 2019

[n eq 0 select 1 else (n+1)*(n+2)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+2,2)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

CoefficientList[Series[(1-x)(1-2x+4x^2)/(1-2x)^3, {x,0,30}], x] (* Michael De Vlieger, Sep 21 2017 *)
Join[{1}, LinearRecurrence[{6,-12,8}, {3,12,40}, 30]] (* G. C. Greubel, Jul 23 2019 *)

vector(30, n, n--; if(n==0,1, 2^(n-1)*binomial(n+2,2) )) \\ G. C. Greubel, Jul 23 2019

[1]+[2^(n-1)*binomial(n+2,2) for n in (1..30)] # G. C. Greubel, Jul 23 2019

G.f.: (1-x)*(1-2*x+4*x^2)/(1-2*x)^3.
For n>0, a(n) = (n+1)*(n+2)*2^(n-2). - Ralf Stephan, Jan 16 2004
a(n) = Sum_{k=0..n} Sum_{i=0..n} (k+1)*binomial(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 7 - 8*log(2).
Sum_{n>=0} (-1)^n/a(n) = 24*log(3/2) - 9. (End)

A229606 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and vertically with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 0, 0, 1, 6, 1, 3, 39, 39, 3, 12, 202, 396, 202, 12, 40, 925, 3040, 3040, 925, 40, 120, 3924, 20714, 35182, 20714, 3924, 120, 336, 15795, 131345, 362100, 362100, 131345, 15795, 336, 896, 61182, 792929, 3476928, 5655616, 3476928, 792929, 61182, 896

Offset: 1

R. H. Hardin, Sep 26 2013

Table starts
...0.....0.......1.........3..........12...........40............120
...0.....6......39.......202.........925.........3924..........15795
...1....39.....396......3040.......20714.......131345.........792929
...3...202....3040.....35182......362100......3476928.......31848813
..12...925...20714....362100.....5655616.....82613904.....1153135492
..40..3924..131345...3476928....82613904...1840258874....39229935270
.120.15795..792929..31848813..1153135492..39229935270..1279020266434
.336.61182.4618048.281845934.15568071652.809714005005.40413033646242

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

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 1 is A052482(n-2).

Empirical for column k:
k=1: a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 6.
k=2: a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) for n > 5.
k=3: a(n) = 15*a(n-1) - 81*a(n-2) + 185*a(n-3) - 162*a(n-4) + 60*a(n-5) - 8*a(n-6) for n > 7.
k=4: [order 9] for n > 11.
k=5: [order 16] for n > 17.
k=6: [order 21] for n > 23.
k=7: [order 46] for n > 47.

A301459 Number of 6-cycles in the n-folded cube graph.

Original entry on oeis.org

0, 0, 96, 320, 3200, 4480, 14336, 43008, 122880, 337920, 901120, 2342912, 5963776, 14909440, 36700160, 89128960, 213909504, 508035072, 1195376640, 2789212160, 6459228160, 14856224768, 33957085184

Offset: 2

Eric W. Weisstein, Mar 21 2018

a(5) is also the number of 6-cycles in the 2-Keller graph.

Eric Weisstein's World of Mathematics, Folded Cube Graph
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Keller Graph
Index entries for linear recurrences with constant coefficients, signature (8, -24, 32, -16).

Cf. A052482 (4-cycles).

Table[Piecewise[{{0, n == 3}, {96, n == 4}, {3200, n == 6}}, 2^(n - 1) n (n - 1) (n - 2)/3], {n, 2, 20}]
Join[{0, 0, 96, 320, 3200}, LinearRecurrence[{8, -24, 32, -16}, {4480, 14336, 43008, 122880, 337920}, 14]]
CoefficientList[Series[32 x^2 (3 - 14 x + 92 x^2 - 516 x^3 + 1456 x^4 - 1920 x^5 + 960 x^6)/(-1 + 2 x)^4, {x, 0, 20}], x]

a(n) = 2^(n - 1)*n*(n - 1)*(n - 2)/3 for n > 6.
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 10.
G.f.: 32*x^4*(3 - 14*x + 92*x^2 - 516*x^3 + 1456*x^4 - 1920*x^5 + 960*x^6)/(-1 + 2*x)^4.

cp's OEIS Frontend

A229637 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally, diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A229694 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A080929 Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).

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A229606 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and vertically with exactly two mistakes, and colors introduced in row-major 0..2 order.

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Examples

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Crossrefs

Formula

A301459 Number of 6-cycles in the n-folded cube graph.

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Programs

Mathematica

Formula

A080929 Sequence associated with a(n) = 2a(n-1) + k(k+2)*a(n-2).