A385601 -id:A385601 - cp's OEIS frontend

A229679 Number of defective 3-colorings of an n X 2 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

0, 2, 36, 360, 2688, 17280, 101376, 559104, 2949120, 15040512, 74711040, 363331584, 1736441856, 8178892800, 38050725888, 175154135040, 798863917056, 3614214979584, 16234976378880, 72464688218112, 321607151124480

Offset: 1

R. H. Hardin, Sep 27 2013

nonn

			Some solutions for n=3:
..0..1....0..1....0..1....0..0....0..1....0..0....0..1....0..1....0..1....0..1
..1..1....1..2....1..0....1..0....1..1....0..1....2..0....1..2....2..2....1..1
..1..2....2..2....2..2....1..1....0..1....2..0....2..2....0..1....2..2....2..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 2 of A229685.

Cf. A385601.

Empirical: a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3) for n>5.
Empirical g.f.: 2*x^2 - 12*x^3*(3-6*x+8*x^2) / (4*x-1)^3. - R. J. Mathar, Sep 29 2013
Empirical: a(n) = 3*2^(2*n-5)*(3 - 5*n + 2*n^2) for n>2. - Colin Barker, Jun 13 2017
From Enrique Navarrete, Jul 08 2025: (Start)
The above empirical formulas are correct.
a(n) = 3*binomial(2*(n-1),2)*2^(2*n-5) for n >= 3.
a(n) = 3*A385601(2*(n-1)) for n >= 3. (End)

A386227 Expansion of e.g.f. cosh(x)^2*(1 + x + x^2/2).

Original entry on oeis.org

1, 1, 3, 6, 20, 40, 152, 224, 1024, 1152, 6272, 5632, 35840, 26624, 194560, 122880, 1015808, 557056, 5144576, 2490368, 25427968, 11010048, 123207680, 48234496, 587202560, 209715200, 2759852032, 905969664, 12817793024, 3892314112, 58921582592, 16642998272, 268435456000, 70866960384

Offset: 0

Enrique Navarrete, Jul 15 2025

a(n) is the number of ternary strings of length n that contain at most two 1's, an even number of 0's, and an even number of 2's.

			a(5) = 40 since the strings are the 30 permutations of 10022, the 5 permutations of 10000, and the 5 permutations of 12222.
a(6) = 152 since the strings are (number of permutations in parentheses): 110022 (90), 110000 (15), 112222 (15), 000022(15), 002222 (15), 222222 (1), 000000 (1).

Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Cf. A229580, A385083, A385601.

a(n) = n*2^(n-2) for odd n >= 3.
a(n) = 2^(n-3)*(binomial(n,2) + 4) for even n >= 4.
a(2n+1) = A229580(n+1).
G.f.: (1 + x - 9*x^2 - 6*x^3 + 32*x^4 + 16*x^5 - 8*x^6 - 32*x^7 - 32*x^8)/((1 - 2*x)^3*(1 + 2*x)^3).

cp's OEIS Frontend

A229679 Number of defective 3-colorings of an n X 2 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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