A001611 -id:A001611 - cp's OEIS frontend

A207661 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 1 0 0 vertically.

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 84, 81, 14, 26, 256, 292, 198, 196, 22, 42, 676, 912, 870, 474, 484, 35, 68, 1764, 2812, 3358, 2774, 1140, 1225, 56, 110, 4624, 8928, 12040, 13902, 9060, 2748, 3136, 90, 178, 12100, 28152, 47320, 55752, 60762, 30440

Offset: 1

R. H. Hardin Feb 19 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..198....870....3358...12040....47320.....182192.....676396.....2611234
.14..196..474...2774...13902...55752...284272....1391724....6009486....29516206
.22..484.1140...9060...60762..264568..1806676...11731102...56370696...366295198
.35.1225.2748..30440..284108.1296732.12327618..111194040..568720390..5097396968
.56.3136.6630.103838.1384420.6454684.87753008.1144562438.5955572036.76412972266

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

R. H. Hardin, Table of n, a(n) for n = 1..420

Column 1 is A001611(n+2)
Column 2 is A207436
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207341

A207752 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 0 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 12, 81, 102, 81, 14, 16, 144, 289, 270, 196, 22, 20, 256, 612, 900, 798, 484, 35, 25, 400, 1296, 2100, 3249, 2354, 1225, 56, 30, 625, 2340, 4900, 8778, 11449, 7210, 3136, 90, 36, 900, 4225, 9450, 23716, 34668, 42436, 22232, 8100, 145, 42

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2....4.....6......9.....12......16......20.......25.......30........36
..4...16....36.....81....144.....256.....400......625......900......1296
..6...36...102....289....612....1296....2340.....4225.....6890.....11236
..9...81...270....900...2100....4900....9450....18225....31185.....53361
.14..196...798...3249...8778...23716...51590...112225...213060....404496
.22..484..2354..11449..34668..104976..247860...585225..1182690...2390116
.35.1225..7210..42436.147084..509796.1341606..3530641..7826035..17347225
.56.3136.22232.157609.617732.2421136.6978660.20115225.47881860.113976976

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

R. H. Hardin, Table of n, a(n) for n = 1..542

Column 1 is A001611(n+2)
Column 2 is A207436
Row 1 is A002620(n+2)
Row 2 is A030179(n+2)
Row 3 is A207118

A207960 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 1 and 1 0 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 102, 81, 14, 19, 169, 281, 297, 196, 22, 28, 361, 699, 989, 932, 484, 35, 41, 784, 1799, 3080, 3923, 2974, 1225, 56, 60, 1681, 4706, 9994, 15839, 15921, 9723, 3136, 90, 88, 3600, 12161, 32358, 66089, 84013, 67007, 32164

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...102....281.....699.....1799......4706......12161.......31356
..9...81...297....989....3080.....9994.....32358.....104176......335940
.14..196...932...3923...15839....66089....274310....1137227.....4717144
.22..484..2974..15921...84013...454223...2439422...13111617....70473286
.35.1225..9723..67007..463482..3256420..22743106..159145878..1113110358
.56.3136.32164.286299.2593815.23710041.215653086.1965912435.17907299730

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

R. H. Hardin, Table of n, a(n) for n = 1..576

Column 1 is A001611(n+2)
Column 2 is A207436
Column 3 is A207495
Row 1 is A000930(n+3)
Row 2 is A207170

A208108 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 78, 81, 14, 25, 225, 169, 171, 196, 22, 40, 625, 611, 361, 406, 484, 35, 64, 1600, 2209, 1805, 841, 990, 1225, 56, 104, 4096, 6016, 9025, 6235, 2025, 2485, 3136, 90, 169, 10816, 16384, 25555, 46225, 22995, 5041, 6328, 8100

Offset: 1

R. H. Hardin Feb 23 2012

Table starts
..2....4....6.....9.....15.......25.......40.......64........104.........169
..4...16...36....81....225......625.....1600.....4096......10816.......28561
..6...36...78...169....611.....2209.....6016....16384......51840......164025
..9...81..171...361...1805.....9025....25555....72361.....292941.....1185921
.14..196..406...841...6235....46225...134160...389376....2189616....12313081
.22..484..990..2025..22995...261121...768544..2262016...18608992...153091129
.35.1225.2485..5041..89815..1600225..4747545.14085009..176312187..2207026441
.56.3136.6328.12769.361261.10220809.30461016.90782784.1769397240.34486347025

			Some solutions for n=10 k=3
..1..0..0....0..0..1....1..0..1....1..1..0....1..0..0....0..1..1....0..0..1
..0..0..1....1..0..0....0..1..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..1..1....0..0..1....1..1..0....1..1..0....0..1..1....1..0..1
..0..1..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..1..1....0..1..1....1..1..0....1..0..0....0..1..1....1..0..1
..0..1..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..0..1....1..1..1....1..1..0....1..0..0....0..1..1....1..1..1
..0..0..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..1..0....0..1..1....1..1..1....1..1..0....1..0..0....0..1..1....0..0..1
..0..0..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..845

Column 1 is A001611(n+2)
Column 2 is A207436
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207730

A081659 a(n) = n + Fibonacci(n+1).

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 19, 28, 42, 64, 99, 155, 245, 390, 624, 1002, 1613, 2601, 4199, 6784, 10966, 17732, 28679, 46391, 75049, 121418, 196444, 317838, 514257, 832069, 1346299, 2178340, 3524610, 5702920, 9227499, 14930387, 24157853, 39088206

Offset: 0

Paul Barry, Mar 26 2003

Row sums of triangle A135222. - Gary W. Adamson, Nov 23 2007

a(n) is the F(n+1)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where F(n) = Fibonacci numbers = A000045(n). - Jaroslav Krizek, Oct 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001611 (first differences), A002062, A135222.

List([0..40], n-> n + Fibonacci(n+1) ); # G. C. Greubel, Nov 20 2019

[n+Fibonacci(n+1): n in [0..40]]; // Vincenzo Librandi, Aug 10 2013

with(combinat); seq(n + fibonacci(n+1), n=0..40); # G. C. Greubel, Nov 20 2019

Table[ Fibonacci[n+1]+n, {n, 0, 38}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)
CoefficientList[Series[(x^3+x-1)/((x-1)^2 (x^2+x-1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)
LinearRecurrence[{3,-2,-1,1},{1,2,4,6},40] (* Harvey P. Dale, Mar 02 2016 *)

numlib::fibonacci(n)+n-1 $ n = 1..48; // Zerinvary Lajos, May 08 2008

a(n)=n+fibonacci(n) \\ Charles R Greathouse IV, Oct 07 2015

[n+fibonacci(n+1) for n in range(40)] # G. C. Greubel, Feb 12 2019

a(n) = (sqrt(5)*(1+sqrt(5))^(n+1) - sqrt(5)*(1-sqrt(5))^(n+1))/(10*2^n) + n.
G.f.: (1-x-x^3)/((1-x-x^2)*(1-x)^2).
From Jaroslav Krizek, Oct 28 2009: (Start)
a(0) = 1, a(n) = a(n-1) + A000045(n-1) + 1 for n >= 1.
a(0) = 1, a(n) = a(n-1) + A000045(n+1) - A000045(n) + 1 for n >= 1.
a(0) = 1, a(1) = 2, a(2) = 4, a(n) = a(n-1) + a(n-2) - (n-3) n >= 3. (End)
E.g.f.: (1/10)*exp(-2*x/(1+sqrt(5)))*(5 - sqrt(5) + (5 + sqrt(5))*exp(sqrt(5)*x) + 10*exp((1/2)*(1+sqrt(5))*x)*x). - Stefano Spezia, Nov 20 2019

A207741 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 0 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 114, 81, 14, 25, 225, 361, 387, 196, 22, 40, 625, 1425, 1849, 1414, 484, 35, 64, 1600, 5625, 10535, 10201, 5302, 1225, 56, 104, 4096, 20550, 60025, 86355, 58081, 20265, 3136, 90, 169, 10816, 75076, 327075, 731025

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2....4.....6.......9.......15.........25..........40............64
..4...16....36......81......225........625........1600..........4096
..6...36...114.....361.....1425.......5625.......20550.........75076
..9...81...387....1849....10535......60025......327075.......1782225
.14..196..1414...10201....86355.....731025.....5959350......48580900
.22..484..5302...58081...733363....9259849...113534330....1392036100
.35.1225.20265..335241..6349893..120275089..2215586241...40813292529
.56.3136.78120.1946025.55343835.1573946929.43590708750.1207251562500

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

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

Column 1 is A001611(n+2)
Column 2 is A207436
Column 3 is A207712
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207427

A207885 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 82, 81, 14, 19, 169, 221, 177, 196, 22, 28, 361, 493, 575, 408, 484, 35, 41, 784, 1095, 1360, 1673, 942, 1225, 56, 60, 1681, 2654, 3106, 4387, 4881, 2233, 3136, 90, 88, 3600, 6203, 8652, 10207, 14225, 14825, 5348, 8100, 145, 129

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2....4....6.....9.....13.....19......28.......41.......60........88.......129
..4...16...36....81....169....361.....784.....1681.....3600......7744.....16641
..6...36...82...221....493...1095....2654.....6203....14182.....33242.....77781
..9...81..177...575...1360...3106....8652....22114....53620....139806....360132
.14..196..408..1673...4387..10207...33976....97989...250976....751786...2196861
.22..484..942..4881..14225..33631..138552...452415..1213630...4316538..14467167
.35.1225.2233.14825..49586.119278..618202..2332602..6535778..28238814.110355600
.56.3136.5348.45411.175747.430317.2857532.12501427.36429646.195734448.894867543

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

R. H. Hardin, Table of n, a(n) for n = 1..578

Column 1 is A001611(n+2)
Column 2 is A207436
Column 3 is A207483
Row 1 is A000930(n+3)
Row 2 is A207170
Row 3 is A207763

A207938 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 0 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 14, 81, 98, 64, 10, 22, 196, 271, 200, 100, 12, 35, 484, 844, 643, 350, 144, 14, 56, 1225, 2706, 2422, 1271, 556, 196, 16, 90, 3136, 8977, 9430, 5594, 2239, 826, 256, 18, 145, 8100, 30168, 38207, 25490, 11256, 3641, 1168, 324, 20, 234

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2...4....6....9....14.....22......35.......56.......90.......145........234
..4..16...36...81...196....484....1225.....3136.....8100.....21025......54756
..6..36...98..271...844...2706....8977....30168...102384....349069....1193648
..8..64..200..643..2422...9430...38207...156792...649758...2703377...11276024
.10.100..350.1271..5594..25490..121313...584386..2841676..13864995...67793828
.12.144..556.2239.11256..58602..319439..1760946..9794226..54631117..305277128
.14.196..826.3641.20568.120276..737575..4570122.28555126.178852957.1121957980
.16.256.1168.5581.34986.226850.1544037.10609482.73474400.509887759.3543126698

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

R. H. Hardin, Table of n, a(n) for n = 1..546

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A207106
Column 4 is A207107
Row 1 is A001611(n+2)
Row 2 is A207436

A208840 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 78, 81, 14, 26, 256, 282, 171, 196, 22, 42, 676, 768, 855, 406, 484, 35, 68, 1764, 2430, 2421, 3010, 990, 1225, 56, 110, 4624, 7086, 9801, 8736, 11242, 2485, 3136, 90, 178, 12100, 21588, 31419, 49126, 33088, 44275

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..171....855...2421.....9801....31419....116919.....394965.....1419849
.14..196..406...3010...8736....49126...169974....833364....3166030....14462714
.22..484..990..11242..33088...272206...992574...6800596...28280758...173714530
.35.1225.2485..44275.131355..1644265..6206445..62470275..277136755..2417186345
.56.3136.6328.179032.533568.10399480.40122936.613538688.2842543480.36689660504

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

R. H. Hardin, Table of n, a(n) for n = 1..1465

Column 1 is A001611(n+2)
Column 2 is A207436
Column 3 is A208103
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A208689

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)=2*a(k-1)+7*a(k-2)-6*a(k-3)
n=5: a(k)=2*a(k-1)+12*a(k-2)-11*a(k-3)
n=6: a(k)=2*a(k-1)+20*a(k-2)-19*a(k-3)
n=7: a(k)=2*a(k-1)+33*a(k-2)-32*a(k-3)

A209650 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 14, 81, 102, 64, 10, 22, 196, 270, 216, 100, 12, 35, 484, 798, 630, 390, 144, 14, 56, 1225, 2354, 2156, 1215, 636, 196, 16, 90, 3136, 7210, 7128, 4690, 2079, 966, 256, 18, 145, 8100, 22232, 24990, 16830, 8904, 3276, 1392, 324, 20, 234

Offset: 1

R. H. Hardin Mar 11 2012

Table starts
..2...4....6....9....14.....22.....35......56.......90......145.......234
..4..16...36...81...196....484...1225....3136.....8100....21025.....54756
..6..36..102..270...798...2354...7210...22232....69570...218950....693810
..8..64..216..630..2156...7128..24990...87136...311040..1112150...4018716
.10.100..390.1215..4690..16830..65765..251160...994050..3911375..15639390
.12.144..636.2079..8904..34012.145775..597856..2579940.10954895..47622744
.14.196..966.3276.15386..61754.287140.1247736..5805450.26247900.122620446
.16.256.1392.4860.24808.103664.518700.2364992.11769120.56106300.279344520

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

R. H. Hardin, Table of n, a(n) for n = 1..2828

Column 2 is A016742
Column 3 is A086113
Row 1 is A001611(n+2)
Row 2 is A207436
Row 3 is A207747

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 2*n^3 + 6*n^2 - 2*n
k=4: a(n) = 9*n^3 + (9/2)*n^2 - (9/2)*n
k=5: a(n) = (7/2)*n^4 + 21*n^3 - (7/2)*n^2 - 7*n
k=6: a(n) = 22*n^4 + (88/3)*n^3 - 22*n^2 - (22/3)*n
k=7: a(n) = 7*n^5 + 70*n^4 + (35/3)*n^3 - (105/2)*n^2 - (7/6)*n

cp's OEIS Frontend

A207661 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 1 0 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207752 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 0 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207960 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 1 and 1 0 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A208108 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 1 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A081659 a(n) = n + Fibonacci(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

MuPAD

PARI

Sage

Formula

A207741 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 0 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207885 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 1 1 and 1 1 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207938 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 0 1 and 1 1 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A208840 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 1 0 vertically.

Views

Author

Keywords

Comments

Examples

Links