A006355 -id:A006355 - cp's OEIS frontend

A208028 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 1 0 1 vertically.

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 102, 81, 13, 26, 256, 378, 279, 169, 19, 42, 676, 1260, 1377, 741, 361, 28, 68, 1764, 4374, 5895, 4823, 1995, 784, 41, 110, 4624, 14946, 26685, 26845, 17119, 5404, 1681, 60, 178, 12100, 51384, 118179, 158847, 123709

Offset: 1

R. H. Hardin Feb 22 2012

Table starts
..2....4.....6.....10......16.......26........42.........68.........110
..4...16....36....100.....256......676......1764.......4624.......12100
..6...36...102....378....1260.....4374.....14946......51384......176238
..9...81...279...1377....5895....26685....118179.....527913.....2350215
.13..169...741...4823...26845...158847....917293....5349227....31070195
.19..361..1995..17119..123709...955073...7184755...54606513...413322903
.28..784..5404..61292..574560..5788524..56728924..561913408..5542832148
.41.1681.14555.218243.2652823.34901455.445530887.5753550295.73969794325

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

R. H. Hardin, Table of n, a(n) for n = 1..763

Column 1 is A000930(n+3)
Column 2 is A207170
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A060521

A208501 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 0 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 60, 81, 13, 26, 256, 144, 126, 169, 19, 42, 676, 324, 324, 234, 361, 28, 68, 1764, 756, 828, 650, 456, 784, 41, 110, 4624, 1728, 2124, 1794, 1406, 896, 1681, 60, 178, 12100, 3996, 5436, 4992, 4104, 3024, 1722, 3600, 88

Offset: 1

R. H. Hardin Feb 27 2012

Table starts
..2....4....6...10....16....26.....42......68.....110......178......288
..4...16...36..100...256...676...1764....4624...12100....31684....82944
..6...36...60..144...324...756...1728....3996....9180....21168....48708
..9...81..126..324...828..2124...5436...13932...35676....91404...234108
.13..169..234..650..1794..4992..13806...38376..106236...295074...817362
.19..361..456.1406..4104.12654..37430..114494..341012..1037552..3103650
.28..784..896.3024..9912.33488.111328..374416.1249472..4192384.14013664
.41.1681.1722.6150.21976.79376.286754.1037300.3752320.13577396.49125954

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

R. H. Hardin, Table of n, a(n) for n = 1..1253

Column 1 is A000930(n+3)
Column 2 is A207170
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207590

A208688 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 78, 81, 14, 26, 256, 282, 189, 196, 21, 42, 676, 768, 927, 490, 441, 31, 68, 1764, 2430, 2889, 3430, 1113, 961, 46, 110, 4624, 7086, 11727, 12096, 11067, 2449, 2116, 68, 178, 12100, 21588, 40581, 66094, 41013, 34627

Offset: 1

R. H. Hardin Mar 01 2012

Table starts
..2....4....6.....10.....16......26.......42........68........110.........178
..4...16...36....100....256.....676.....1764......4624......12100.......31684
..6...36...78....282....768....2430.....7086.....21588......64230......193554
..9...81..189....927...2889...11727....40581....154359.....554733.....2062215
.14..196..490...3430..12096...66094...269766...1331988....5795314....27403166
.21..441.1113..11067..41013..301035..1346961...8556723...42184905...249260739
.31..961.2449..34627.133207.1332721..6398617..53340739..290904031..2188890625
.46.2116.5474.111642.444912.6219706.31733422.358035204.2130519946.21086588370

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

R. H. Hardin, Table of n, a(n) for n = 1..1360

Column 1 is A038718(n+2)
Column 2 is A207069
Column 3 is A207724
Row 1 is A006355(n+2)
Row 2 is A206981

Empirical for row n:
n=1: a(k)=a(k-1)+a(k-2)
n=2: a(k)=2*a(k-1)+2*a(k-2)-a(k-3)
n=3: a(k)=2*a(k-1)+4*a(k-2)-3*a(k-3)
n=4: a(k)=a(k-1)+10*a(k-2)+2*a(k-3)-10*a(k-4)
n=5: a(k)=a(k-1)+17*a(k-2)+4*a(k-3)-32*a(k-4)
n=6: a(k)=a(k-1)+26*a(k-2)+6*a(k-3)-78*a(k-4)
n=7: a(k)=a(k-1)+39*a(k-2)+9*a(k-3)-180*a(k-4)

A221573 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 0, 2, 0, 5, 2, 0, 10, 9, 4, 0, 17, 26, 25, 6, 0, 26, 59, 100, 57, 10, 0, 37, 114, 289, 342, 141, 16, 0, 50, 197, 676, 1293, 1210, 345, 26, 0, 65, 314, 1369, 3734, 5913, 4240, 853, 42, 0, 82, 471, 2500, 8991, 20944, 26911, 14898, 2097, 68, 0, 101, 674, 4225, 19014

Offset: 1

R. H. Hardin Jan 20 2013

Table starts
...0.....0.......0........0.........0..........0...........0............0
...2.....5......10.......17........26.........37..........50...........65
...2.....9......26.......59.......114........197.........314..........471
...4....25.....100......289.......676.......1369........2500.........4225
...6....57.....342.....1293......3734.......8991.......19014........36497
..10...141....1210.....5913.....20944......59705......145800.......317233
..16...345....4240....26911....117104.....395641.....1116400......2754635
..26...853...14898...122621....655198....2622817.....8550512.....23923281
..42..2097...52306...558547...3665306...17385993....65485386....207761745
..68..5149..183684..2544357..20505052..115249117...501533796...1804315029
.110.12633..645006.11590169.114711980..763966685..3841097940..15669633131
.178.31013.2264978.52796369.641737294.5064207645.29417832750.136083460405

			Some solutions for n=6 k=4
..2....2....3....1....0....0....2....1....2....2....3....1....4....3....1....4
..0....0....0....1....4....0....2....3....0....4....0....3....4....3....3....4
..1....1....4....3....4....2....0....4....0....0....4....2....3....0....1....2
..3....1....2....1....3....3....2....4....0....3....3....2....1....3....0....0
..3....4....1....1....0....0....0....3....1....2....1....4....0....3....3....0
..0....1....4....1....3....0....4....0....3....4....3....4....2....3....0....0

R. H. Hardin, Table of n, a(n) for n = 1..2080

Column 1 is A006355
Row 2 is A002522
Row 4 is A082044

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 3*a(n-1) -2*a(n-2) +4*a(n-4)
k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
k=4: a(n) = 5*a(n-1) -3*a(n-2) +a(n-3) +15*a(n-4) +3*a(n-5) for n>6
k=5: a(n) = 5*a(n-1) +3*a(n-2) +9*a(n-4) +6*a(n-5) +3*a(n-6)
k=6: a(n) = 7*a(n-1) -4*a(n-2) +6*a(n-3) +26*a(n-4) +10*a(n-5) +16*a(n-6) +12*a(n-8)
k=7: a(n) = 7*a(n-1) +4*a(n-2) +5*a(n-3) +20*a(n-4) +20*a(n-5) +23*a(n-6) -6*a(n-7) +3*a(n-8)
Empirical for row n:
n=2: a(n) = 1*n^2 + 1
n=3: a(n) = 1*n^3 - 1*n^2 + 3*n - 1
n=4: a(n) = 1*n^4 + 2*n^2 + 1
n=5: a(n) = 1*n^5 + 1*n^4 - 2*n^3 + 12*n^2 - 15*n + 9 for n>2
n=6: a(n) = 1*n^6 + 2*n^5 - 5*n^4 + 24*n^3 - 41*n^2 + 50*n - 31 for n>3
n=7: a(n) = 1*n^7 + 3*n^6 - 7*n^5 + 29*n^4 - 41*n^3 + 45*n^2 - 33*n + 19 for n>2

A052995 Expansion of 2x(1 - x)/(1 - 3*x + x^2).

Original entry on oeis.org

0, 2, 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314, 150050, 392836, 1028458, 2692538, 7049156, 18454930, 48315634, 126491972, 331160282, 866988874, 2269806340, 5942430146, 15557484098, 40730022148, 106632582346, 279167724890, 730870592324

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Terms >=4 give solutions x to floor(phi^2*x^2) - floor(phi*x)^2 = 5, where phi =(1 + sqrt(5))/2. - Benoit Cloitre, Mar 16 2003
Except for the first term, positive values of x (or y) satisfying x^2 - 18*x*y + y^2 + 256 = 0. - Colin Barker, Feb 14 2014
a(n+1) is the square of the distance AB, where A is the point (F(n), F(n+1)), B is the 90-degree rotation of A about the origin, and F(n)=A000045(n) are the Fibonacci numbers. - Burak Muslu, Mar 24 2021

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 30.
B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 60-61.

Colin Barker, Table of n, a(n) for n = 0..1000
Younseok Choo, Some Results on the Infinite Sums of Reciprocal Generalized Fibonacci Numbers, International Journal of Mathematical Analysis, 12(12) (2018), 621-629.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1072.
Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Bisection of A006355.
First differences of A025169.
Cf. A000032, A000045, A055819, A006355, A025169.

spec := [S, S=Prod(Sequence(Union(Prod(Sequence(Z),Z),Z)),Union(Z,Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

LinearRecurrence[{3, -1}, {0, 2, 4}, 30] (* or *)
Nest[Append[#, 3 #[[-1]] - #[[-2]]] &, {0, 2, 4}, 27] (* or *)
CoefficientList[Series[-2 x (-1 + x)/(1 - 3 x + x^2), {x, 0, 29}], x] (* Michael De Vlieger, Jul 18 2018 *)

concat(0, Vec(2*x*(1-x)/(1-3*x+x^2) + O(x^50))) \\ Colin Barker, Mar 30 2016

a(n) = fibonacci(max(0,2*n-1))<<1; \\ Kevin Ryde, Mar 25 2021

G.f.: -2*x*(-1 + x)/(1 - 3*x + x^2).
a(0) = 0, a(1) = 2, a(2) = 4; for n > 0, a(n) - 3*a(n+1) + a(n+2) = 0.
a(n) = A069403(n-1)+1.
a(n) = Sum(2/5*(-1 + 4*_alpha)*_alpha^(-1-n), _alpha = RootOf(_Z^2 - 3*_Z + 1)).
a(n) = 2*Fibonacci(2*n-1) = 2*A001519(n) for n > 0. - Vladeta Jovovic, Mar 19 2003
a(n+2) = F(n)^2 + F(n+3)^2 = 2*F(n+1)^2 + 2*F(n+2)^2, where F = A000045. -  N. J. A. Sloane, Feb 20 2005
a(n) = 1/2*(F(2*n+8) mod F(2*n+2)) for n > 2. - Gary Detlefs, Nov 22 2010
a(n) = F(n-3)*F(n-1) + F(n)*F(n+2) for n > 0, F(-2) = -1, F(-1) = 1. - Bruno Berselli, Nov 03 2015
a(n) = (2^(-n)*((3 - sqrt(5))^n*(1 + sqrt(5)) + (-1 + sqrt(5))*(3 + sqrt(5))^n))/sqrt(5) for n > 0. - Colin Barker, Mar 30 2016
a(n) = Fibonacci(2*n-2) + Lucas(2*n-2) for n > 0. - Bruno Berselli, Oct 13 2017
a(n) = Lucas(2*n) - Fibonacci(2*n) for n > 0. - Diego Rattaggi, Mar 08 2023

More terms from James Sellers, Jun 05 2000

A207774 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 84, 81, 14, 26, 256, 292, 192, 196, 21, 42, 676, 912, 828, 450, 441, 31, 68, 1764, 2812, 3130, 2514, 972, 961, 46, 110, 4624, 8928, 11230, 11950, 7164, 2040, 2116, 68, 178, 12100, 28152, 43260, 49122, 43264, 20104

Offset: 1

R. H. Hardin Feb 20 2012

Table starts
..2....4....6....10.....16......26.......42........68........110.........178
..4...16...36...100....256.....676.....1764......4624......12100.......31684
..6...36...84...292....912....2812.....8928.....28152......87972......277292
..9...81..192...828...3130...11230....43260....163710.....604340.....2295512
.14..196..450..2514..11950...49122...240346...1128862....4877154....23327018
.21..441..972..7164..43264..195160..1246206...7381202...35627804...220514498
.31..961.2040.20104.157472..759482..6513576..49975640..259077468..2150453548
.46.2116.4278.57458.597090.3023026.35632160.364308938.1972417684.22562076620

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

R. H. Hardin, Table of n, a(n) for n = 1..513

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207341
Row 4 is A207342

A207808 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 10, 16, 100, 102, 100, 16, 26, 256, 378, 370, 256, 26, 42, 676, 1260, 1970, 1232, 676, 42, 68, 1764, 4374, 9040, 9168, 4238, 1764, 68, 110, 4624, 14946, 43990, 57184, 44538, 14406, 4624, 110, 178, 12100, 51384, 209050, 382288

Offset: 1

R. H. Hardin Feb 20 2012

Table starts
..2....4.....6......10.......16........26.........42..........68...........110
..4...16....36.....100......256.......676.......1764........4624.........12100
..6...36...102.....378.....1260......4374......14946.......51384........176238
.10..100...370....1970.....9040.....43990.....209050.....1002960.......4793390
.16..256..1232....9168....57184....382288....2485392....16340928.....106947696
.26..676..4238...44538...379444...3511534...31431114...285153752....2572767886
.42.1764.14406..212814..2472540..31569510..388134978..4844843724...60105117534
.68.4624.49164.1022652.16206848.285774964.4829044276.82999241712.1416863447084

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

R. H. Hardin, Table of n, a(n) for n = 1..650

Column 1 is A006355(n+2)
Column 2 is A206981
Column 3 is A207249
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A060521

A207858 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 15, 81, 102, 100, 16, 25, 225, 289, 370, 256, 26, 40, 625, 1071, 1369, 1232, 676, 42, 64, 1600, 3969, 7289, 5929, 4238, 1764, 68, 104, 4096, 13230, 38809, 44121, 26569, 14406, 4624, 110, 169, 10816, 44100, 178088, 328329

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2....4.....6......9.......15........25.........40..........64...........104
..4...16....36.....81......225.......625.......1600........4096.........10816
..6...36...102....289.....1071......3969......13230.......44100........153090
.10..100...370...1369.....7289.....38809.....178088......817216.......3976696
.16..256..1232...5929....44121....328329....2047902....12773476......85393582
.26..676..4238..26569...279219...2934369...24999522...212984836....1971051046
.42.1764.14406.117649..1737981..25674489..298294290..3465676900...44249929850
.68.4624.49164.522729.10873197.226171521.3584335104.56804048896.1001624438528

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

R. H. Hardin, Table of n, a(n) for n = 1..391

Column 1 is A006355(n+2)
Column 2 is A206981
Column 3 is A207249
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207704

A207895 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 0 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 13, 81, 84, 100, 16, 19, 169, 198, 292, 256, 26, 28, 361, 462, 870, 912, 676, 42, 41, 784, 1080, 2446, 3358, 2812, 1764, 68, 60, 1681, 2520, 6952, 11196, 12040, 8928, 4624, 110, 88, 3600, 5886, 20168, 38160, 49442, 47320, 28152

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2....4.....6......9......13......19.......28........41.........60.........88
..4...16....36.....81.....169.....361......784......1681.......3600.......7744
..6...36....84....198.....462....1080.....2520......5886......13746......32100
.10..100...292....870....2446....6952....20168.....57838.....164914.....473632
.16..256...912...3358...11196...38160...135714....471374....1605156....5575548
.26..676..2812..12040...49442..205258...856480...3559040...14795830...61589756
.42.1764..8928..47320..231900.1153532..5888458..29719004..148308700..747498440
.68.4624.28152.182192.1070030.6420722.39828558.243318702.1460399404.8907998820

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

R. H. Hardin, Table of n, a(n) for n = 1..419

Column 1 is A006355(n+2)
Column 2 is A206981
Column 3 is A207341
Column 4 is A207662
Row 1 is A000930(n+3)
Row 2 is A207170

A207918 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 1 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 13, 81, 98, 100, 16, 19, 169, 271, 358, 256, 26, 28, 361, 665, 1309, 1152, 676, 42, 41, 784, 1675, 4181, 5371, 3910, 1764, 68, 60, 1681, 4344, 13759, 21145, 23637, 12994, 4624, 110, 88, 3600, 11081, 46800, 86255, 117835

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2....4.....6......9......13.......19........28.........41..........60
..4...16....36.....81.....169......361.......784.......1681........3600
..6...36....98....271.....665.....1675......4344......11081.......28136
.10..100...358...1309....4181....13759.....46800.....156135......518564
.16..256..1152...5371...21145....86255....366330....1520815.....6276388
.26..676..3910..23637..117835...612439...3327954...17621905....92785236
.42.1764.12994.101069..628945..4105063..28188778..187980955..1245595210
.68.4624.43596.438103.3426491.28280693.247024548.2088382007.17532981294

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

R. H. Hardin, Table of n, a(n) for n = 1..418

Column 1 is A006355(n+2)
Column 2 is A206981
Column 3 is A207462
Row 1 is A000930(n+3)
Row 2 is A207170
Row 3 is A207306

cp's OEIS Frontend

A208028 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 1 0 1 vertically.

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A208501 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 0 and 1 1 1 vertically.

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A208688 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 0 1 vertically.

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A221573 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by something other than 1.

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A052995 Expansion of 2*x*(1 - x)/(1 - 3*x + x^2).

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A207774 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 0 1 0 vertically.

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A207808 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 0 and 1 0 1 vertically.

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A207858 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 0 and 1 0 1 vertically.

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A207895 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 0 and 1 1 1 vertically.

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A052995 Expansion of 2x(1 - x)/(1 - 3*x + x^2).