A028447 -id:A028447 - cp's OEIS frontend

A181206 T(n,k) = number of n X k matrices containing a permutation of 1..n*k moving each element at most to a neighboring position.

1, 2, 2, 3, 9, 3, 5, 32, 32, 5, 8, 121, 229, 121, 8, 13, 450, 1845, 1845, 450, 13, 21, 1681, 14320, 32000, 14320, 1681, 21, 34, 6272, 112485, 535229, 535229, 112485, 6272, 34, 55, 23409, 880163, 9049169, 19114420, 9049169, 880163, 23409, 55, 89, 87362

Offset: 1

R. H. Hardin, Oct 10 2010

Also, the number of perfect matchings in the graph P_2 X P_k X P_n. - Andrew Howroyd, May 17 2017

			Table starts:
..1......2.........3............5................8..................13
..2......9........32..........121..............450................1681
..3.....32.......229.........1845............14320..............112485
..5....121......1845........32000...........535229.............9049169
..8....450.....14320.......535229.........19114420...........692276437
.13...1681....112485......9049169........692276437.........53786626921
.21...6272....880163....152526845......24972353440.......4161756233501
.34..23409...6895792...2573281769.....901990734650.....322462050747008
.55..87362..54003765..43402320448...32567565264292...24976513162427653
.89.326041.422983905.732106008249.1176040842289105.1934824269280528177
...
All solutions for 3X2
..1..2....1..2....1..2....1..2....1..2....1..2....1..2....1..2....1..2....1..4
..3..4....4..3....4..3....4..6....3..4....3..6....5..4....5..3....5..6....3..2
..5..6....5..6....6..5....3..5....6..5....5..4....3..6....6..4....3..4....5..6
...
..1..4....1..4....2..1....2..1....2..1....2..1....2..1....2..1....2..1....2..1
..3..2....5..2....4..3....4..3....4..6....3..4....3..4....3..6....5..4....5..3
..6..5....3..6....5..6....6..5....3..5....5..6....6..5....5..4....3..6....6..4
...
..2..1....2..4....2..4....2..4....3..1....3..1....3..1....3..2....3..2....3..2
..5..6....1..3....1..3....1..6....4..2....4..2....5..2....1..4....1..4....1..6
..3..4....5..6....6..5....3..5....5..6....6..5....6..4....5..6....6..5....5..4
...
..3..4....3..4
..1..2....1..2
..5..6....6..5

Alois P. Heinz, Table of n, a(n) for n = 1..210 (first 180 terms from R. H. Hardin)

Columns k=1-10 give: A000045(n+1), A006253, A028447, A028448, A028449, A028450, A028451, A287052, A287053, A287054.

Main diagonal gives A181205.

A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)(x^3+2x^2+x-1) ).

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 36, 81, 169, 361, 784, 1681, 3600, 7744, 16641, 35721, 76729, 164836, 354025, 760384, 1633284, 3508129, 7535025, 16184529, 34762816, 74666881, 160376896, 344473600, 739894401, 1589218225, 3413480625, 7331811876, 15747991081, 33825095056

Offset: 0

R. J. Mathar, Dec 06 2013

a(n) is the number of tilings of a 3 X 2 X n room with bricks of 1 X 1 X 3 shape (and in that respect a generalization of A028447 which fills 3 X 2 X n rooms with bricks of 1 X 1 X 2 shape).
The inverse INVERT transform is 1, 0, 3, 2, 2, 4, 4, 6, 8, 10, .. , continued as in A068924.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) with half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2;3)-combs. A (w,g;m)-comb is a tile composed of m pieces of dimensions w X 1 separated horizontally by gaps of width g. - Michael A. Allen, Sep 24 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: counts derived from transfer matrices, arXiv:1406.7788 [math.CO], 2014, see eq. (39).
Index entries for linear recurrences with constant coefficients, signature (1,1,3,1,-1,-1).

Cf. A000930.

A233247 := proc(n)
    A000930(n)^2 ;
end proc:
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <1|0|1>>^n)[3, 3]^2:
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2013

Table[Sum[Binomial[n-2i, i], {i,0,n/3}]^2, {n,0,50}] (* Wesley Ivan Hurt, Dec 06 2013 *)
LinearRecurrence[{1,1,3,1,-1,-1},{1,1,1,4,9,16},40] (* Harvey P. Dale, Jan 14 2015 *)
CoefficientList[Series[(1-x^3-x^2)/((x^3-x^2-1)*(x^3+2*x^2+x-1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 29 2017 *)

my(x='x+O('x^50)); Vec((1-x^3-x^2)/((x^3-x^2-1)*(x^3+2*x^2+x-1))) \\ G. C. Greubel, Apr 29 2017

a(n) = A000930(n)^2.

a(n) = a(n-1) + a(n-3) + 2*Sum_{r=3..n} ( A000931(r+2)*a(n-r) ). - Michael A. Allen, Sep 24 2024

cp's OEIS Frontend

A181206 T(n,k) = number of n X k matrices containing a permutation of 1..n*k moving each element at most to a neighboring position.

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A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)(x^3+2x^2+x-1) ).

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cp's OEIS Frontend

A181206 T(n,k) = number of n X k matrices containing a permutation of 1..n*k moving each element at most to a neighboring position.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)*(x^3+2*x^2+x-1) ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)(x^3+2x^2+x-1) ).