A183719 -id:A183719 - cp's OEIS frontend

A183712 1/20 of the number of (n+1) X 3 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

5, 17, 54, 174, 559, 1797, 5776, 18566, 59677, 191821, 616574, 1981866, 6370351, 20476345, 65817520, 211558554, 680016837, 2185791545, 7025832918, 22583273462, 72589861759, 233327025821, 749987665760, 2410700161342, 7748761123965, 24906995867477

Offset: 1

R. H. Hardin, Jan 06 2011

Column 2 of A183719. [Corrected by M. F. Hasler, Oct 07 2014]
This sequence counts closed walks of length (n+2) at the vertex of a triangle, to which a loop has been added to one of the remaining vertices and two loops has been added to the third vertex. - David Neil McGrath, Sep 04 2014
From Greg Dresden, Mar 02 2025: (Start)
a(n) is the number of ways to tile, with squares and dominoes, a 2 X n board with two extra spaces at the end. Here is the board for n=3:
    ___
   |||_| 
   |||_|||,
and here is one of the a(3)=54 possible tilings of this board:
    ___
   |_| |_
   |||_|_|.
Compare to A033505 (tilings of 2 X n board with one extra space at the end) and A030186 (tilings of 2 X n board with no extra spaces at the end). (End)

			Some solutions for 5 X 3:
..0..1..4....1..2..0....4..0..4....4..3..4....4..0..4....1..4..0....3..4..2
..3..2..3....0..3..4....2..1..3....0..2..0....3..2..3....2..3..1....1..0..1
..4..1..0....1..2..1....4..0..4....4..3..4....0..1..0....0..4..0....2..4..3
..3..2..3....0..3..4....3..2..3....0..2..1....4..2..3....1..3..1....1..0..1
..4..0..4....1..2..1....4..1..0....4..3..0....0..1..0....0..4..0....2..3..2
...
...R..L.......R..L.......R..L.......L..R.......R..L.......L..R.......R..L...
...L..R.......L..R.......L..R.......R..L.......L..R.......R..L.......L..R...
...R..L.......R..L.......R..L.......L..R.......R..L.......L..R.......R..L...
...L..R.......L..R.......L..R.......R..L.......L..R.......R..L.......L..R...

R. H. Hardin, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (3,1,-1).

Cf. A183727, A183719, A030186, A033505.

a(n)=([0,1,1;1,1,1;1,1,2]^(n+2))[1,1] \\ Charles R Greathouse IV, Sep 15 2014

Vec(x*(5+2*x-2*x^2)/(1-3*x-x^2+x^3) + O(x^50)) \\ Colin Barker, Mar 16 2016

a(n) = 3*a(n-1) + a(n-2) - a(n-3).
The top left element of A^(n+2) where A=(0,1,1;1,1,1;1,1,2). - David Neil McGrath, Sep 04 2014
a(n) ~ c*k^n where k = 1.629316... is the largest root of x^3 - 3x^2 - x + 1 and c = 1.6293... is conjecturally the largest root of 148x^3 - 296x^2 + 90x - 1. - Charles R Greathouse IV, Sep 15 2014
G.f.: x*(5+2*x-2*x^2) / (1-3*x-x^2+x^3). - Colin Barker, Mar 16 2016
a(n) = A030186(n) + A033505(n). - Greg Dresden, Mar 02 2025

A183713 1/20 of the number of (n+1) X 4 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

Original entry on oeis.org

12, 54, 224, 950, 4012, 16964, 71712, 303170, 1281664, 5418314, 22906232, 96837444, 409385940, 1730703022, 7316648160, 30931557950, 130764969444, 552816553732, 2337064300200, 9880075964922, 41768598769664, 176579193270290

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Column 3 of A183719.

			Some solutions for 3 X 4:
..0..4..1..4....0..4..0..4....4..0..4..0....3..0..3..4....3..2..3..2
..2..3..2..3....1..2..1..3....3..2..3..2....2..1..2..0....0..1..0..1
..0..4..1..0....0..3..0..4....0..1..4..0....3..4..3..4....3..2..4..2
...
...L..R..L.......L..R..L.......R..L..R.......R..L..R.......L..R..L...
...R..L..R.......R..L..R.......L..R..L.......L..R..L.......R..L..R...

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A183719.

Empirical: a(n) = 2*a(n-1) + 10*a(n-2) - 10*a(n-4) - 2*a(n-5) + a(n-6).

Empirical g.f.: 2*x*(6 + 15*x - 2*x^2 - 19*x^3 - 4*x^4 + 2*x^5) / ((1 - x)*(1 + x)*(1 - 2*x - 9*x^2 - 2*x^3 + x^4)). - Colin Barker, Apr 04 2018

A183714 1/20 of the number of (n+1) X 5 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

Original entry on oeis.org

29, 174, 950, 5362, 30113, 169560, 954496, 5374440, 30261345, 170394226, 959449106, 5402445266, 30419982145, 171288302164, 964487232600, 5430818438104, 30579761182213, 172188006484798, 969553341271046

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Column 4 of A183719.

			Some solutions for 3 X 5:
..2..1..2..0..1....4..3..0..3..4....3..4..3..4..2....0..3..4..3..4
..3..0..3..4..3....0..2..1..2..1....2..1..2..0..1....1..2..0..1..0
..2..1..2..0..2....4..3..4..3..4....4..0..3..4..2....0..3..4..2..3
...
...L..R..L..R.......L..R..L..R.......R..L..R..L.......L..R..L..R...
...R..L..R..L.......R..L..R..L.......L..R..L..R.......R..L..R..L...

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 6*a(n-1) + 5*a(n-2) - 42*a(n-3) - 2*a(n-4) + 84*a(n-5) - 20*a(n-6) - 46*a(n-7) + 19*a(n-8) + 2*a(n-9) - a(n-10).

A183715 1/20 of the number of (n+1) X 6 0..4 arrays with every 2X2 subblock strictly increasing clockwise or counterclockwise with one decrease.

Original entry on oeis.org

70, 559, 4012, 30113, 224640, 1683197, 12606120, 94463507, 707826798, 5304230928, 39748015308, 297860692491, 2232084261366, 16726639262465, 125344918169856, 939301219166473, 7038871324696794, 52747415646208987, 395274995490521924

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Column 5 of A183719.

			Some solutions for 3 X 6:
..3..1..3..2..3..1....0..2..0..1..4..0....2..3..1..2..1..2....1..0..2..1..2..1
..4..0..4..1..4..0....4..3..4..2..3..1....0..4..0..3..0..4....3..4..3..4..3..4
..2..1..3..2..3..2....1..2..0..1..4..0....2..3..1..2..1..2....2..1..2..0..2..1
...
...L..R..L..R..L.......R..L..R..L..R.......R..L..R..L..R.......L..R..L..R..L...
...R..L..R..L..R.......L..R..L..R..L.......L..R..L..R..L.......R..L..R..L..R...

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=2*a(n-1)+54*a(n-2)-2*a(n-3)-738*a(n-4)-312*a(n-5)+4078*a(n-6)+2098*a(n-7)-10802*a(n-8)-4874*a(n-9)+14854*a(n-10)+4874*a(n-11)-10802*a(n-12)-2098*a(n-13)+4078*a(n-14)+312*a(n-15)-738*a(n-16)+2*a(n-17)+54*a(n-18)-2*a(n-19)-a(n-20).

A183717 1/20 of the number of (n+1) X 8 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

Original entry on oeis.org

408, 5776, 71712, 954496, 12606120, 168070828, 2239265280, 29887741084, 398877017736, 5325395568832, 71097798240000, 949288848361824, 12674717188992600, 169233802374938432, 2259618796412155200

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Column 7 of A183719.

			Some solutions for 3 X 8:
..0..4..0..4..0..4..0..4....1..0..2..0..1..0..1..0....3..0..3..0..4..0..4..0
..2..3..1..2..1..2..1..2....3..4..3..4..2..4..2..3....2..1..2..1..3..1..3..2
..0..4..0..4..0..3..0..3....2..0..1..0..1..0..1..0....3..4..3..0..4..0..4..1
...
...L..R..L..R..L..R..L.......L..R..L..R..L..R..L.......R..L..R..L..R..L..R...
...R..L..R..L..R..L..R.......R..L..R..L..R..L..R.......L..R..L..R..L..R..L...

R. H. Hardin, Table of n, a(n) for n = 1..200

A183718 1/20 of the number of (n+1) X 9 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

Original entry on oeis.org

985, 18566, 303170, 5374440, 94463507, 1680902120, 29887741084, 532844962952, 9498797049301, 169444214222538, 3022597001362206, 53927539247910656, 962146234750661827, 17166919088214504572, 306297928172424966016

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Column 8 of A183719.

			Some solutions for 3 X 9:
..0..2..0..1..0..2..0..2..1....0..3..0..4..0..3..4..3..0
..4..3..4..2..4..3..4..3..0....1..2..1..2..1..2..1..2..1
..1..2..0..1..0..2..0..2..1....0..3..4..3..0..3..0..3..4
...
...R..L..R..L..R..L..R..L.......L..R..L..R..L..R..L..R...
...L..R..L..R..L..R..L..R.......R..L..R..L..R..L..R..L...

R. H. Hardin, Table of n, a(n) for n = 1..156

A183711 1/20 of the number of (n+1) X (n+1) 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

Original entry on oeis.org

2, 17, 224, 5362, 224640, 16823812, 2239265280, 532844962952, 226169896329216, 171735135418632976, 232966965220983791616, 565591167958003196615968, 2455196078535760615522467840

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Diagonal of A183719.

			Some solutions for 3 X 3:
..0..1..0....3..2..3....2..3..1....4..3..0....4..0..4....4..0..3....0..1..0
..3..2..3....4..0..4....1..4..0....1..2..1....2..1..2....2..1..2....4..3..4
..4..1..0....3..2..3....2..3..1....0..3..0....3..0..3....3..4..3....1..2..1
...
...R..L.......L..R.......R..L.......L..R.......R..L.......R..L.......R..L...
...L..R.......R..L.......L..R.......R..L.......L..R.......L..R.......L..R...

A183716 1/20 of the number of (n+1) X 7 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

Original entry on oeis.org

169, 1797, 16964, 169560, 1683197, 16823812, 168070828, 1680902120, 16810325640, 168150768145, 1681983501886, 16825267289824, 168307255896111, 1683632068657357, 16841927176565292, 168475631338846804

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Column 6 of A183719.

			Some solutions for 3 X 7:
..4..0..4..0..4..0..4....1..0..1..4..1..4..1....2..4..2..4..2..4..2
..3..2..3..2..3..1..3....2..3..2..3..2..3..2....1..0..1..0..1..0..1
..0..1..4..1..4..0..4....1..4..1..4..0..4..0....2..4..2..4..2..4..2
...
...R..L..R..L..R..L.......L..R..L..R..L..R.......R..L..R..L..R..L...
...L..R..L..R..L..R.......R..L..R..L..R..L.......L..R..L..R..L..R...

Empirical: a(n)=12*a(n-1)+41*a(n-2)-731*a(n-3)-236*a(n-4)+17377*a(n-5)-11599*a(n-6)-210068*a(n-7)+239143*a(n-8)+1435333*a(n-9)-2045028*a(n-10)-5829087*a(n-11)+9632462*a(n-12)+14361840*a(n-13)-27309714*a(n-14)-21401634*a(n-15)+48701640*a(n-16)+18436650*a(n-17)-56085498*a(n-18)-7403128*a(n-19)+42308694*a(n-20)-1027934*a(n-21)-20976128*a(n-22)+2642614*a(n-23)+6772235*a(n-24)-1305140*a(n-25)-1387863*a(n-26)+326941*a(n-27)+171884*a(n-28)-44771*a(n-29)-11807*a(n-30)+3188*a(n-31)+387*a(n-32)-103*a(n-33)-4*a(n-34)+a(n-35).

cp's OEIS Frontend

A183712 1/20 of the number of (n+1) X 3 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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A183713 1/20 of the number of (n+1) X 4 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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A183714 1/20 of the number of (n+1) X 5 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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A183715 1/20 of the number of (n+1) X 6 0..4 arrays with every 2X2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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A183717 1/20 of the number of (n+1) X 8 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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A183718 1/20 of the number of (n+1) X 9 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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A183711 1/20 of the number of (n+1) X (n+1) 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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A183716 1/20 of the number of (n+1) X 7 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

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