A265232 -id:A265232 - cp's OEIS frontend

A265233 Number of 3 X n arrays containing n copies of 0..2 with no equal vertical neighbors and new values introduced sequentially from 0.

1, 1, 7, 56, 495, 4686, 46456, 475392, 4976271, 52977890, 571434402, 6228357312, 68468597544, 758063599632, 8443936740960, 94545206802816, 1063391499647631, 12007844534804202, 136068111377744686, 1546682224461979920, 17630279034262961010, 201470426310372260580

Offset: 0

R. H. Hardin, Dec 06 2015

nonn

Row 3 of A265232.

			Some solutions for n=4
..0..1..0..2....0..1..2..2....0..1..0..0....0..1..1..2....0..1..1..2
..2..0..2..0....2..0..1..0....2..2..2..2....1..2..2..0....2..0..0..0
..1..1..1..2....0..1..2..1....1..1..0..1....0..0..1..2....1..1..2..2

R. H. Hardin, Table of n, a(n) for n = 0..106

Cf. A265232.

Conjecture: n^2*a(n) +(-19*n^2+19*n-6)*a(n-1) +96*(n-1)^2*a(n-2) -144*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 08 2015
Conjecture: a(n) ~ 2^(2*n - 1) * 3^(n - 1/2) / (Pi*n). - Vaclav Kotesovec, Mar 08 2023
If conjectured recurrence is true then ogf = (hypergeom([1/3,2/3],[1],27*x*(4*x-1)^2)+5)/6. - Mark van Hoeij, Nov 28 2024

a(0)=1 prepended by Alois P. Heinz, Nov 28 2024

A265234 Number of 4 X n arrays containing n copies of 0..4-1 with no equal vertical neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 43, 2592, 184740, 14439456, 1196114464, 103142395392, 9160513923648, 832211576040960, 76971887847571968, 7223525356855099392, 686117529041422350336, 65834293657115919826944, 6371837299781950752276480

Offset: 1

R. H. Hardin, Dec 06 2015

nonn

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

Manuel Kauers and Christoph Koutschan, Table of n, a(n) for n = 1..495 (terms 1..31 from R. H. Hardin).
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Row 4 of A265232.

From Manuel Kauers and Christoph Koutschan, Mar 01 2023: (Start)
a(n) = coefficient of x^n*y^n*z^n in (1/24)*(2*x^2 + 6*x*y + 6*x^2*y + 2*y^2 + 6*x*y^2 + 2*x^2*y^2 + 6*x*z + 6*x^2*z + 6*y*z + 24*x*y*z + 6*x^2*y*z + 6*y^2*z + 6*x*y^2*z + 2*z^2 + 6*x*z^2 + 2*x^2*z^2 + 6*y*z^2 + 6*x*y*z^2 + 2*y^2*z^2)^n.
Recurrence of order 6 and degree 6: 5*(n + 5)*(832*n^2 + 5785*n + 8460)*(n + 6)^3*a(n + 6) - 4*(n + 5)*(126464*n^5 + 2941016*n^4 + 26840735*n^3 + 119399663*n^2 + 256228730*n + 208319000)*a(n + 5) + 16*(310336*n^6 + 7680621*n^5 + 78610375*n^4 + 426421788*n^3 + 1294537774*n^2 + 2087600280*n + 1398239904)*a(n + 4) + 128*(n + 4)*(1161472*n^5 + 24822356*n^4 + 207271023*n^3 + 841828441*n^2 + 1653171497*n + 1242989235)*a(n + 3) - 768*(n + 3)*(n + 4)*(3709888*n^4 + 58438003*n^3 + 333112832*n^2 + 813878537*n + 716118600)*a(n + 2) + 9216*(n + 2)*(n + 3)*(n + 4)*(1743872*n^3 + 20496944*n^2 + 74692297*n + 84692065)*a(n + 1) - 34836480*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(832*n^2 + 7449*n + 15077)*a(n) = 0. (End)
a(n) ~ 2^(2*n - 19/2) * 3^(3*n + 7/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

A265229 Number of n X 2 arrays containing 2 copies of 0..n-1 with no equal vertical neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 2, 7, 43, 372, 4027, 51871, 773186, 13083385, 247698481, 5186925696, 119023766737, 2969884019977, 80056947698498, 2318432654628847, 71785166633148187, 2366425763631216756, 82748313392542136011

Offset: 1

R. H. Hardin, Dec 06 2015

			Some solutions for n=4:
..0..1....0..1....0..0....0..0....0..1....0..1....0..1....0..1....0..1....0..1
..2..0....2..3....1..2....1..1....2..3....2..3....2..0....2..2....2..3....2..3
..3..2....1..2....3..1....2..2....3..1....3..0....1..3....0..1....3..0....1..0
..1..3....3..0....2..3....3..3....0..2....1..2....3..2....3..3....2..1....3..2

Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.

Column 2 of A265232.

A265230 Number of nX3 arrays containing 3 copies of 0..n-1 with no equal vertical neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 4, 56, 2592, 222850, 29357385, 5460708807, 1360701035812, 437196618018421, 175924945354032520

Offset: 1

R. H. Hardin, Dec 06 2015

nonn

Column 3 of A265232.

			Some solutions for n=4
..0..0..1....0..0..1....0..0..1....0..1..2....0..1..2....0..0..1....0..1..1
..2..1..3....2..3..2....1..2..3....3..0..1....3..0..0....2..2..0....2..3..0
..3..3..2....1..2..1....3..0..2....1..3..3....2..2..1....1..3..3....1..0..3
..1..2..0....3..0..3....2..3..1....2..2..0....3..1..3....3..2..1....3..2..2

Cf. A265232.

A265231 Number of nX4 arrays of permutations of 4 copies of 0..n-1 with no equal vertical neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 8, 495, 184740, 164933040, 277778385204, 785263649477535

Offset: 1

R. H. Hardin, Dec 06 2015

nonn

Column 4 of A265232.

			Some solutions for n=4
..0..1..1..2....0..0..1..1....0..0..1..0....0..1..0..2....0..0..1..1
..1..0..0..3....2..2..0..3....2..2..3..3....3..2..3..3....2..3..0..0
..2..3..3..2....3..3..3..2....1..3..1..2....2..3..0..1....1..2..3..1
..3..2..0..1....2..0..1..1....3..2..0..1....1..2..1..0....3..3..2..2

Cf. A265232.

A265235 Number of 5Xn arrays containing n copies of 0..5-1 with no equal vertical neighbors and new values introduced sequentially from 0.

Original entry on oeis.org

1, 372, 222850, 164933040, 137409486772, 123531282436770, 117115132044401100, 115473661661749763520, 117339977328886229407120, 122120444042051198450868012, 129586955944964431527460363320

Offset: 1

R. H. Hardin, Dec 06 2015

nonn

Row 5 of A265232.

			Some solutions for n=2
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..3....2..0....2..3....2..3....2..3....2..0....2..3....2..3....2..3....2..3
..0..1....3..1....0..1....4..1....1..4....3..2....4..4....4..1....4..2....0..4
..4..4....4..4....4..4....0..2....4..3....4..4....1..3....3..4....1..4....4..1
..2..3....2..3....3..2....4..3....0..2....1..3....2..0....2..0....3..0....3..2

Cf. A265232.

cp's OEIS Frontend

A265233 Number of 3 X n arrays containing n copies of 0..2 with no equal vertical neighbors and new values introduced sequentially from 0.

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A265234 Number of 4 X n arrays containing n copies of 0..4-1 with no equal vertical neighbors and new values introduced sequentially from 0.

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A265229 Number of n X 2 arrays containing 2 copies of 0..n-1 with no equal vertical neighbors and new values introduced sequentially from 0.

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A265230 Number of nX3 arrays containing 3 copies of 0..n-1 with no equal vertical neighbors and new values introduced sequentially from 0.

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A265231 Number of nX4 arrays of permutations of 4 copies of 0..n-1 with no equal vertical neighbors and new values introduced sequentially from 0.

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A265235 Number of 5Xn arrays containing n copies of 0..5-1 with no equal vertical neighbors and new values introduced sequentially from 0.

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