A159288 -id:A159288 - cp's OEIS frontend

A240284 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

1, 2, 2, 2, 8, 3, 4, 19, 19, 4, 4, 76, 80, 38, 7, 8, 181, 570, 262, 114, 10, 8, 741, 2574, 3457, 1461, 251, 15, 16, 1779, 20764, 28654, 33183, 5443, 612, 24, 16, 7308, 97348, 443168, 484146, 218658, 24490, 1656, 35, 32, 17561, 802835, 3980245, 13490093, 5646644

Offset: 1

R. H. Hardin, Apr 03 2014

Table starts
..1....2.......2.........4............4.............8..............8
..2....8......19........76..........181...........741...........1779
..3...19......80.......570.........2574.........20764..........97348
..4...38.....262......3457........28654........443168........3980245
..7..114....1461.....33183.......484146......13490093......224906182
.10..251....5443....218658......5646644.....281488213.....8597299482
.15..612...24490...1851080.....88953626....8199368365...463717321235
.24.1656..117962..15760838...1357879302..225885684501.23104690637116
.35.3758..459193.110599613..17350066110.5291794810655
.54.9630.2147788.945852472.272318368893

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A159288

Row 1 is A016116

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 15]
Empirical for row n:
n=1: a(n) = 2*a(n-2)
n=2: a(n) = 12*a(n-2) -24*a(n-4) +31*a(n-6) -16*a(n-8)
n=3: [order 48] for n>51

A241397 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

Original entry on oeis.org

2, 2, 3, 4, 5, 4, 6, 9, 13, 7, 8, 23, 29, 28, 10, 14, 44, 85, 97, 64, 15, 20, 93, 201, 480, 340, 142, 24, 30, 204, 689, 1657, 2780, 1156, 318, 35, 48, 368, 1929, 8697, 15339, 17211, 4068, 726, 54, 70, 761, 4068, 31654, 129985, 160947, 102782, 14763, 1634, 83, 108

Offset: 1

R. H. Hardin, Apr 20 2014

Table starts
..2....2......4........6..........8..........14..........20.........30
..3....5......9.......23.........44..........93.........204........368
..4...13.....29.......85........201.........689........1929.......4068
..7...28.....97......480.......1657........8697.......31654......92204
.10...64....340.....2780......15339......129985......667050....2949778
.15..142...1156....17211.....160947.....2234804....18589398..134190457
.24..318...4068...102782....1622867....39781828...541660724.7518340973
.35..726..14763...645484...18627122...821631365.21331876978
.54.1634..52950..3936420..196990531.16108803423
.83.3695.190950.24633252.2356216195

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

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A159288(n+1)

Row 1 is A239851

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 38]
Empirical for row n:
n=1: a(n) = a(n-2) +2*a(n-3)
n=2: [order 22] for n>23

A241054 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or zero plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

Original entry on oeis.org

2, 3, 2, 4, 3, 4, 7, 2, 4, 6, 10, 10, 3, 6, 8, 15, 18, 24, 6, 8, 14, 24, 18, 60, 64, 6, 12, 20, 35, 46, 93, 163, 132, 15, 13, 30, 54, 58, 297, 280, 598, 690, 31, 20, 48, 83, 102, 507, 1423, 1392, 3411, 2142, 58, 28, 70, 124, 173, 1264, 4167, 10921, 13273, 11283, 7144, 170, 38

Offset: 1

R. H. Hardin, Apr 15 2014

Table starts
..2..3...4.....7......10........15..........24..........35..........54
..2..3...2....10......18........18..........46..........58.........102
..4..4...3....24......60........93.........297.........507........1264
..6..6...6....64.....163.......280........1423........4167.......13389
..8..8...6...132.....598......1392.......10921.......72769......370453
.14.12..15...690....3411.....13273......189680.....2667280....18820225
.20.13..31..2142...11283.....89910.....1511923....30914092...376386754
.30.20..58..7144...72578...1128052....35582068..1432670661.26960360814
.48.28.170.30662..404421..13331118...776191453.62057946683
.70.38.388.95669.2220973.128026529.14877945554

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

R. H. Hardin, Table of n, a(n) for n = 1..143

Column 1 is A239851

Row 1 is A159288(n+1)

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: a(n) = 2*a(n-2) -a(n-4) +a(n-5) -a(n-7) +a(n-8) +a(n-11) for n>15
Empirical for row n:
n=1: a(n) = a(n-2) +2*a(n-3)
n=2: [order 15] for n>17
n=3: [order 70] for n>85

A159287 Expansion of x^2/(1-x^2-2*x^3).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 4, 5, 6, 13, 16, 25, 42, 57, 92, 141, 206, 325, 488, 737, 1138, 1713, 2612, 3989, 6038, 9213, 14016, 21289, 32442, 49321, 75020, 114205, 173662, 264245, 402072, 611569, 930562, 1415713, 2153700, 3276837, 4985126, 7584237, 11538800

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj').
From Greg Dresden, Nov 15 2024: (Start)
a(n) is the number of ways to tile a 2 X (n+1) board with L-shaped trominos and S-shaped quadrominos, where the first tile must be an upright L. For example, here are the a(7)=4 ways to tile a 2 X 8 board:
  ._____________.   ._____________.
  | |  |  |  | |   | |  |  |  | |
  |_|_||__|___|   |_|___|||___|
  ._____________.   ._____________.
  | |  | |  |  |   | |  |  | |  |
  |_|_|_|___||   |__|___||__|_| (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier.
Yüksel Soykan, A Study on Generalized Jacobsthal-Padovan Numbers, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 227-251.
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159285, A159286, A159288.

Essentially the same as A052947.

I:=[0,0,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{0, 1, 2}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 24 2011 *)
CoefficientList[Series[x^2/(1-x^2-2x^3),{x,0,50}],x] (* Harvey P. Dale, May 29 2021 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[0;0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x^2/(1-x^2-2*x^3).
a(n) = A052947(n-2). - R. J. Mathar, Nov 10 2009
a(n) = a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, May 23 2023
From Greg Dresden, Nov 17 2024: (Start)
a(2*n+1) = 2*a(n)^2 + 2*a(n+1)*a(n+2).
a(3*n+1) = Sum_{i=1..n} a(3*i-2)*2^(n-i). (End)

A240792 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or one plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest or zero plus the sum of the elements antidiagonally to its northeast, modulo 4.

Original entry on oeis.org

2, 3, 3, 4, 4, 4, 7, 5, 5, 7, 10, 10, 8, 10, 10, 15, 14, 19, 27, 13, 15, 24, 24, 34, 37, 49, 14, 24, 35, 53, 82, 132, 85, 50, 30, 35, 54, 70, 278, 552, 460, 142, 89, 32, 54, 83, 140, 669, 2277, 2009, 1130, 386, 115, 36, 83, 124, 237, 1969, 11588, 15611, 6993, 6393, 979, 182

Offset: 1

R. H. Hardin, Apr 12 2014

Table starts
..2..3...4....7.....10......15.........24.........35..........54.........83
..3..4...5...10.....14......24.........53.........70.........140........237
..4..5...8...19.....34......82........278........669........1969.......3553
..7.10..27...37....132.....552.......2277......11588.......35799.....109054
.10.13..49...85....460....2009......15611......90831......522104....2386511
.15.14..50..142...1130....6993......92681.....983289.....7075546...64616034
.24.30..89..386...6393...69732....1691066...33021579...408482271.5524873850
.35.32.115..979..22255..398128...16946464..651434319.13673101597
.54.36.182.1988..62705.1646410..117913446.8338435721
.83.67.289.5128.252760.9472037.1022475213

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

R. H. Hardin, Table of n, a(n) for n = 1..142

Row and column 1 are A159288(n+1)

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 17] for n>19
k=3: [order 76] for n>87
Empirical for row n:
n=1: a(n) = a(n-2) +2*a(n-3)
n=2: [order 17] for n>21

A240376 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or one plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

2, 3, 3, 4, 11, 4, 7, 21, 21, 7, 10, 67, 75, 67, 10, 15, 155, 450, 450, 155, 15, 24, 353, 1729, 5161, 1729, 353, 24, 35, 998, 7233, 36398, 36398, 7233, 998, 35, 54, 2256, 36148, 271764, 486179, 271764, 36148, 2256, 54, 83, 5639, 139855, 2492182, 6436979

Offset: 1

R. H. Hardin, Apr 04 2014

Table starts
..2.....3.......4..........7...........10............15.............24
..3....11......21.........67..........155...........353............998
..4....21......75........450.........1729..........7233..........36148
..7....67.....450.......5161........36398........271764........2492182
.10...155....1729......36398.......486179.......6436979......110122847
.15...353....7233.....271764......6436979.....169838571.....5145071133
.24...998...36148....2492182....110122847....5145071133...296413962369
.35..2256..139855...17380978...1403151574..121626491919.12947745036751
.54..5639..645733..143489019..20729739995.3384817934297
.83.14624.2919837.1182292032.314460587672

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A159288(n+1)

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 15] for n>17

A240406 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 14, 14, 4, 7, 24, 77, 24, 7, 10, 77, 235, 235, 77, 10, 15, 182, 1381, 1304, 1381, 182, 15, 24, 397, 5566, 13648, 13648, 5566, 397, 24, 35, 1164, 21413, 98837, 257243, 98837, 21413, 1164, 35, 54, 2626, 114951, 692520, 3366314, 3366314, 692520

Offset: 1

R. H. Hardin, Apr 04 2014

Table starts
..1....2.......3.........4............7............10............15
..2....5......14........24...........77...........182...........397
..3...14......77.......235.........1381..........5566.........21413
..4...24.....235......1304........13648.........98837........692520
..7...77....1381.....13648.......257243.......3366314......43820839
.10..182....5566.....98837......3366314......80283637....1935005419
.15..397...21413....692520.....43820839....1935005419...85908009949
.24.1164..114951...6686074....766539340...62076655091.5128168982319
.35.2626..447479..47001423...9754760605.1450984782112
.54.6439.1942846.368795251.139442821024

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

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A159288

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 15]

A159285 Expansion of (1+3x)/(1-x^2-2x^3).

Original entry on oeis.org

1, 3, 1, 5, 7, 7, 17, 21, 31, 55, 73, 117, 183, 263, 417, 629, 943, 1463, 2201, 3349, 5127, 7751, 11825, 18005, 27327, 41655, 63337, 96309, 146647, 222983, 339265, 516277, 785231, 1194807, 1817785, 2765269, 4207399, 6400839, 9737937, 14815637

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj')

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link]
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159286, A159287, A159288.

I:=[1,3,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{0, 1, 2}, {1, 3, 1}, 50] (* G. C. Greubel, Jun 27 2018 *)
CoefficientList[Series[(1+3x)/(1-x^2-2x^3),{x,0,40}],x] (* Harvey P. Dale, Jan 21 2019 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;3;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A052947(n) + 3*A052947(n-1). - R. J. Mathar, Mar 23 2023

A159286 Expansion of (x-1)^2/(1-x^2-2*x^3).

Original entry on oeis.org

1, -2, 2, 0, -2, 4, -2, 0, 6, -4, 6, 8, -2, 20, 14, 16, 54, 44, 86, 152, 174, 324, 478, 672, 1126, 1628, 2470, 3880, 5726, 8820, 13486, 20272, 31126, 47244, 71670, 109496, 166158, 252836, 385150, 585152

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj').

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link].
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159285, A159287, A159288.

I:=[1, -2, 2]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

CoefficientList[Series[(x-1)^2/(1-x^2-2*x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,2},{1,-2,2},40]  (* Harvey P. Dale, Apr 24 2011 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;-2;2])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A159288(n) - 3*A159287(n+1). - R. J. Mathar, Apr 10 2009
a(1)=1, a(2)=-2, a(3)=2, a(n) = 1*a(n-2) + 2*a(n-3) for n >= 3. - Harvey P. Dale, Apr 24 2011
a(n) = A078026(n)-A078026(n-1). - R. J. Mathar, Mar 23 2023

cp's OEIS Frontend

A240284 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

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A241397 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

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A241054 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or zero plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

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A159287 Expansion of x^2/(1-x^2-2*x^3).

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Magma

Mathematica

PARI

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A240792 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or one plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest or zero plus the sum of the elements antidiagonally to its northeast, modulo 4.

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A240376 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or one plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

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Formula

A240406 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

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A159285 Expansion of (1+3*x)/(1-x^2-2*x^3).

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Author

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Programs

Magma

Mathematica

PARI

Formula

A159286 Expansion of (x-1)^2/(1-x^2-2*x^3).

A159285 Expansion of (1+3x)/(1-x^2-2x^3).