A159288 -id:A159288 - cp's OEIS frontend

A159284 Expansion of x(1+x)/(1-x^2-2x^3).

0, 1, 1, 1, 3, 3, 5, 9, 11, 19, 29, 41, 67, 99, 149, 233, 347, 531, 813, 1225, 1875, 2851, 4325, 6601, 10027, 15251, 23229, 35305, 53731, 81763, 124341, 189225, 287867, 437907, 666317, 1013641, 1542131, 2346275, 3569413, 5430537

Offset: 0

Creighton Dement, Apr 08 2009

a(n) is the number of composition of n+1 into parts congruent to 0 or 2 modulo 3. - Joerg Arndt, Apr 21 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Matthias Beck and Neville Robbins, Variations on a Generatingfunctional Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence, arXiv:1403.0665 [math.NT], 2014.
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159285, A159286, A159287, A159288.

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

CoefficientList[Series[x (1+x)/(1-x^2-2x^3),{x,0,50}],x] (* or *) LinearRecurrence[ {0,1,2},{0,1,1},50] (* Harvey P. Dale, Jul 16 2013 *)

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

a(n) = abs(A078028(n-1)). - R. J. Mathar, Jul 05 2012
a(n) = a(n-2) + 2*a(n-3), a(0)=0, a(1) = a(2) =1. - G. C. Greubel, Apr 30 2017
a(n) = A052947(n-1)+A052947(n-2). - R. J. Mathar, Mar 23 2023

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A241255 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 one 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, 2, 4, 5, 4, 7, 4, 17, 6, 10, 10, 13, 39, 8, 15, 12, 34, 47, 87, 14, 24, 22, 71, 120, 174, 212, 20, 35, 41, 135, 446, 545, 606, 488, 30, 54, 59, 356, 1202, 3404, 2570, 2111, 1134, 48, 83, 120, 734, 3822, 11700, 25190, 13328, 6647, 2644, 70, 124, 171, 1705, 11428

Offset: 1

R. H. Hardin, Apr 18 2014

Table starts
..2....3.....4.......7........10.........15.........24.........35.........54
..2....5.....4......10........12.........22.........41.........59........120
..4...17....13......34........71........135........356........734.......1705
..6...39....47.....120.......446.......1202.......3822......11428......35540
..8...87...174.....545......3404......11700......50281.....252069.....959723
.14..212...606....2570.....25190.....124372.....752717....6264519...34987493
.20..488..2111...13328....225191....1558957...13138271..205823368.1596727720
.30.1134..6647...70264...2057343...22016913..265444281.8317272277
.48.2644.21752..390840..20539926..362503120.6509404451
.70.6118.70595.2166393.204332167.6317232175

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

R. H. Hardin, Table of n, a(n) for n = 1..127

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: [order 34] for n>37
Empirical for row n:
n=1: a(n) = a(n-2) +2*a(n-3)
n=2: [order 10] for n>12
n=3: [order 52] for n>59

A241435 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 one plus the sum of the elements antidiagonally to its northeast, modulo 4.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 7, 10, 2, 7, 10, 21, 22, 3, 10, 15, 45, 74, 97, 5, 15, 24, 88, 158, 515, 213, 6, 24, 35, 181, 448, 1563, 1527, 381, 9, 35, 54, 378, 1272, 5915, 7495, 5304, 1005, 10, 54, 83, 710, 3284, 22712, 45139, 37148, 20690, 1900, 15, 83, 124, 1460, 8331, 76145

Offset: 1

R. H. Hardin, Apr 22 2014

Table starts
..2..3.....4......7.......10.........15..........24..........35..........54
..3..5....10.....21.......45.........88.........181.........378.........710
..4..2....22.....74......158........448........1272........3284........8331
..7..3....97....515.....1563.......5915.......22712.......76145......270960
.10..5...213...1527.....7495......45139......282527.....1304136.....6927135
.15..6...381...5304....37148.....377314.....4122537....29425635...269064197
.24..9..1005..20690...218885....4136727....79901137..1058201862.16609740868
.35.10..1900..61348..1059975...34541924..1231102996.34796682706
.54.15..4137.257119..7257895..418453966.26880310825
.83.21.10518.920918.44141371.4088141292

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

R. H. Hardin, Table of n, a(n) for n = 1..126

Column and row 1 are A159288(n+1)

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 11] for n>13
Empirical for row n:
n=1: a(n) = a(n-2) +2*a(n-3)
n=2: [order 22] for n>24

A239986 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or two plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 13, 16, 7, 1, 6, 22, 56, 40, 10, 1, 7, 38, 171, 261, 84, 15, 1, 8, 65, 530, 1391, 935, 208, 24, 1, 9, 107, 1495, 7113, 9079, 4113, 474, 35, 1, 10, 169, 4059, 31226, 83658, 70107, 16724, 1047, 54, 1, 11, 257, 10121, 131242, 652346

Offset: 1

R. H. Hardin, Mar 30 2014

Table starts
..1....1......1........1..........1............1............1............1
..2....3......4........5..........6............7............8............9
..3....6.....13.......22.........38...........65..........107..........169
..4...16.....56......171........530.........1495.........4059........10121
..7...40....261.....1391.......7113........31226.......131242.......514539
.10...84....935.....9079......83658.......652346......4803152.....33097266
.15..208...4113....70107....1174822.....16721012....226886115...2823199343
.24..474..16724...514297...15307425....381369904...9004871354.198719581101
.35.1047..63746..3533132..192702130...9009351655.404795616742
.54.2530.275188.27478686.2733573580.233083355837

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

R. H. Hardin, Table of n, a(n) for n = 1..128

Column 1 is A159288

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: a(n) = 2*a(n-2) +10*a(n-3) -a(n-4) -5*a(n-5) -15*a(n-6) +a(n-7) +4*a(n-8) +2*a(n-9) +10*a(n-10) +5*a(n-11) -6*a(n-13)
Empirical for row n:
n=1: a(n) = 1
n=2: a(n) = n + 1
n=3: a(n) = (1/24)*n^4 - (1/4)*n^3 + (71/24)*n^2 - (43/4)*n + 23 for n>3
n=4: [polynomial of degree 10] for n>12
n=5: [polynomial of degree 24] for n>31
n=6: [polynomial of degree 55] for n>73

A241356 T(n,k) = Number of n X k 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 one 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, 2, 3, 4, 3, 4, 6, 9, 3, 7, 8, 17, 19, 4, 10, 14, 23, 51, 55, 5, 15, 20, 53, 61, 128, 72, 5, 24, 30, 103, 230, 228, 248, 124, 7, 35, 48, 160, 641, 1721, 615, 624, 243, 8, 54, 70, 344, 960, 5663, 6307, 2062, 1323, 370, 9, 83, 108, 643, 3746, 11909, 32942, 35880, 6380, 2715

Offset: 1

R. H. Hardin, Apr 20 2014

Table starts
..2..2...4.....6......8.......14........20.........30.........48.........70
..3..3...9....17.....23.......53.......103........160........344........643
..4..3..19....51.....61......230.......641........960.......3746.......9339
..7..4..55...128....228.....1721......5663......11909......69946.....220363
.10..5..72...248....615.....6307.....32942......81541.....704210....3476469
.15..5.124...624...2062....35880....247664.....921726...10840453...85630246
.24..7.243..1323...6380...183400...1904754...10693549..198803445.2384535274
.35..8.370..2715..17325...763750..12340892..109041097.3042023002
.54..9.695..5798..60671..4110488.104529676.1490516896
.83.12.956.11469.174659.18352240.729080777

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

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

Column 1 is A159288(n+1).
Column 2 is A226503(n+8).
Row 1 is A239851.

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3).
k=2: a(n) = a(n-3) +a(n-5).
k=3: [order 68] for n > 85.
Empirical for row n:
n=1: a(n) = a(n-2) +2*a(n-3).
n=2: [order 17] for n > 20.

A240153 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 one 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, 5, 4, 7, 8, 11, 7, 10, 19, 20, 25, 10, 15, 36, 107, 67, 62, 15, 24, 57, 186, 676, 254, 144, 24, 35, 120, 450, 1328, 3993, 825, 329, 35, 54, 218, 1641, 5701, 11742, 22412, 2667, 775, 54, 83, 377, 2788, 24419, 88214, 108201, 131005, 8652, 1781, 83, 124, 758

Offset: 1

R. H. Hardin, Apr 02 2014

Table starts
..2....3.....4........7.........10..........15..........24.........35
..3....5.....8.......19.........36..........57.........120........218
..4...11....20......107........186.........450........1641.......2788
..7...25....67......676.......1328........5701.......24419......52676
.10...62...254.....3993......11742.......88214......536744....1652584
.15..144...825....22412.....108201.....1608415....12812998...70360014
.24..329..2667...131005....1056699....30542203...417799464.3795213209
.35..775..8652...731518...10526838...584983851.14482686630
.54.1781.27929..4144347..107361960.11643589675
.83.4150.91436.23263202.1116331018

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

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

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 34] for n>36
Empirical for row n:
n=1: a(n) = a(n-2) +2*a(n-3)
n=2: [order 17] for n>19

A240192 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 three plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 8, 12, 4, 1, 14, 37, 27, 7, 1, 26, 129, 138, 73, 10, 1, 50, 478, 771, 680, 154, 15, 1, 98, 1908, 5240, 7170, 2413, 358, 24, 1, 194, 7868, 40765, 91879, 44594, 10017, 872, 35, 1, 386, 32888, 336257, 1399773, 1005029, 333607, 43956, 1871, 54, 1

Offset: 1

R. H. Hardin, Apr 02 2014

Table starts
..1....1......1.........1...........1............1.............1.............1
..2....5......8........14..........26...........50............98...........194
..3...12.....37.......129.........478.........1908..........7868.........32888
..4...27....138.......771........5240........40765........336257.......2843914
..7...73....680......7170.......91879......1399773......22849697.....385366572
.10..154...2413.....44594.....1005029.....28061567.....865984451...28244997476
.15..358..10017....333607....14022582....733907809...43398047802.2752449791995
.24..872..43956...2715035...206345434..19388521135.2070573220929
.35.1871.159668..17332017..2336659626.394134037392
.54.4438.681760.134735700.33576330306

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

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

Column 1 is A159288

Row 2 is A164094(n-2)

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 13]
Empirical for row n:
n=1: a(n) = a(n-1)
n=2: a(n) = 3*a(n-1) -2*a(n-2) for n>3
n=3: a(n) = 9*a(n-1) -27*a(n-2) +29*a(n-3) +6*a(n-4) -32*a(n-5) +16*a(n-6) for n>9
n=4: [order 29] for n>34

A241283 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 one 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, 4, 4, 3, 10, 7, 5, 9, 24, 10, 13, 41, 36, 56, 15, 17, 126, 236, 139, 132, 24, 35, 224, 773, 1615, 532, 312, 35, 90, 934, 1800, 6783, 12356, 2111, 736, 54, 141, 2741, 16843, 20717, 77955, 96171, 8473, 1736, 83, 288, 5225, 54167, 451318, 309657, 1009773

Offset: 1

R. H. Hardin, Apr 18 2014

Table starts
....2......3........4..........7..........10............15...........24
....4......3........5.........13..........17............35...........90
...10......9.......41........126.........224...........934.........2741
...24.....36......236........773........1800.........16843........54167
...56....139.....1615.......6783.......20717........451318......1543713
..132....532....12356......77955......309657......15616828.....60486552
..312...2111....96171....1009773.....5423164.....730435588...2949336562
..736...8473...761754...14440961...113305157...40083129145.193899487190
.1736..34053..6079503..217830879..2759021846.2390612565177
.4096.136880.48655224.3381893022.75062814060

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

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

Column 1 is A052912

Row 1 is A159288(n+1)

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-3)
k=2: [order 31]
Empirical for row n:
n=1: a(n)=a(n-2)+2*a(n-3)
n=2: [order 17] for n>18

A239599 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 one 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, 4, 3, 10, 3, 4, 24, 15, 4, 7, 56, 64, 31, 4, 10, 132, 244, 187, 82, 5, 15, 312, 1030, 1310, 643, 177, 7, 24, 736, 4303, 9806, 8773, 1737, 458, 8, 35, 1736, 17923, 76769, 128347, 38824, 7461, 1071, 11, 54, 4096, 75264, 611126, 1991329, 1031560, 282333, 24946

Offset: 1

R. H. Hardin, Mar 22 2014

Table starts
..2..4....10.....24.......56........132.........312.........736........1736
..3..3....15.....64......244.......1030........4303.......17923.......75264
..4..4....31....187.....1310.......9806.......76769......611126.....4929897
..7..4....82....643.....8773.....128347.....1991329....31686730...510551001
.10..5...177...1737....38824....1031560....30460289...944810972.30050150163
.15..7...458...7461...282333...12509870...681399736.40483561185
.24..8..1071..24946..1583770..120511910.11135057785
.35.11..2150..78667..8002162.1104844380
.54.12..5209.313003.56967196
.83.16.11204.946740

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

R. H. Hardin, Table of n, a(n) for n = 1..95

Column 1 is A159288(n+1)

Row 1 is A052912

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: a(n) = a(n-2) +a(n-5) for n>6
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-3)
n=2: [order 44] for n>47

A240271 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 one plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

2, 4, 3, 10, 7, 4, 24, 35, 14, 7, 56, 157, 118, 36, 10, 132, 713, 919, 582, 72, 15, 312, 3263, 7562, 8265, 2000, 170, 24, 736, 14895, 64721, 126286, 49921, 8353, 411, 35, 1736, 68101, 563496, 2059061, 1363144, 382690, 37422, 879, 54, 4096, 311509, 4956889

Offset: 1

R. H. Hardin, Apr 03 2014

Table starts
..2....4......10.........24..........56..........132...........312
..3....7......35........157.........713.........3263.........14895
..4...14.....118........919........7562........64721........563496
..7...36.....582.......8265......126286......2059061......34514871
.10...72....2000......49921.....1363144.....40760821....1277623744
.15..170....8353.....382690....19210586...1063706501...63085436203
.24..411...37422....3076452...278945445..27923918285.3004792552569
.35..879..135463...19781372..3200032085.576407548906
.54.2106..580528..154994425.46095401280
.83.4874.2403439.1144262410

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

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A159288(n+1)

Row 1 is A052912

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 13]
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-3)
n=2: [order 26] for n>28

cp's OEIS Frontend

A159284 Expansion of x*(1+x)/(1-x^2-2*x^3).

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A241255 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 one 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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A241435 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 one plus the sum of the elements antidiagonally to its northeast, modulo 4.

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A239986 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or two plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4.

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A240153 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 one 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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A240192 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 three plus the sum of the elements diagonally to its northwest, modulo 4.

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A241283 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 one 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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A239599 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 one 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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A159284 Expansion of x(1+x)/(1-x^2-2x^3).