A006498 -id:A006498 - cp's OEIS frontend

A105774 A "fractal" transform of the Fibonacci numbers: a(1)=1; then if F(n) < k <= F(n+1), a(k) = F(n+1) - a(k - F(n)) where F(n) = A000045(n).

1, 1, 2, 4, 4, 7, 7, 6, 12, 12, 11, 9, 9, 20, 20, 19, 17, 17, 14, 14, 15, 33, 33, 32, 30, 30, 27, 27, 28, 22, 22, 23, 25, 25, 54, 54, 53, 51, 51, 48, 48, 49, 43, 43, 44, 46, 46, 35, 35, 36, 38, 38, 41, 41, 40, 88, 88, 87, 85, 85, 82, 82, 83, 77, 77, 78, 80, 80, 69, 69, 70, 72, 72

Offset: 1

Benoit Cloitre, May 04 2005

nonn

Let tau = (1+sqrt(5))/2; then the missing numbers 3,5,8,10,13,16,18,21,... are given by round(tau^2*k) for k > 0 (A004937).
Indices n such that a(n) = a(n+1) are given by floor(tau^2*k) - 1 for k > 0 (A003622).
Numbers n such that a(n) differs from a(n+1) are given by floor(tau*k+1/tau) for k > 0 (A022342).
Indices n giving isolated terms (a(n) differs from a(n-1) and a(n+1)) are given by floor(tau*floor(tau^2*k)) for k > 0 (A003623).
Remove 0's from the first differences of sorted values; then you get a version of the infinite Fibonacci word (A001468). I.e., sorted values are 1,1,2,4,4,6,7,7,9,9,11,12,12,..., first differences are 0,1,2,0,2,1,0,2,0,2,1,0,2,0,1,...; removing 0's gives 1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,... #{ k : a(k)=k}=infty.

			For 1 = F(2) < k <= F(3) = 2 the rule gives a(2) = 2 - a(1) = 1 ... if 5 = F(5) < k <= F(6) = 8 the rule forces a(6) = 8 - a(6-5) = 8 - a(1) = 7; a(7) = 8 - a(2) = 7; a(8) = 8 - a(3) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10946
Benoit Cloitre and Jeffrey Shallit, Some Fibonacci-Related Sequences, arXiv:2312.11706 [math.CO], 2023.
Jeffrey Shallit, Using automata to prove theorems about sequences, One FLAT World Seminar, 2024.

Cf. A000045, A001468, A003622, A003623, A004937, A006498, A022342, A072649, A085002, A105669, A105670, A105672, A093347, A093348, A283766.

a(A000045(n)) = A006498(n-1) for n >= 1. - Typo corrected by Antti Karttunen, Mar 17 2017
limsup a(n)/n = tau and liminf a(n)/n = (tau+2)/5 where tau = (1+sqrt(5))/2. - Corrected by Jeffrey Shallit, Dec 17 2023
a(n) mod 2 = A085002(n) - Benoit Cloitre, May 10 2005
a(1) = 1; for n > 1, a(n) = A000045(2+A072649(n-1)) - a(n-A000045(1 + A072649(n-1))). - Antti Karttunen, Mar 17 2017

A207599 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 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, 15, 81, 90, 64, 10, 25, 225, 225, 168, 100, 12, 40, 625, 825, 441, 270, 144, 14, 64, 1600, 3025, 1995, 729, 396, 196, 16, 104, 4096, 9240, 9025, 3915, 1089, 546, 256, 18, 169, 10816, 28224, 30400, 21025, 6765, 1521, 720, 324, 20, 273

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2...4...6....9....15.....25.....40......64......104......169.......273
..4..16..36...81...225....625...1600....4096....10816....28561.....74529
..6..36..90..225...825...3025...9240...28224....93912...312481....997815
..8..64.168..441..1995...9025..30400..102400...403520..1590121...5746377
.10.100.270..729..3915..21025..75400..270400..1223560..5536609..21791133
.12.144.396.1089..6765..42025.157440..589824..3005184.15311569..64177113
.14.196.546.1521.10725..75625.292600.1132096..6404216.36228361.159389139
.16.256.720.2025.15975.126025.499840.1982464.12318592.76545001.350003745

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

R. H. Hardin, Table of n, a(n) for n = 1..1250

Column 2 is A016742.
Column 3 is A152746.
Column 4 is A016946(n-1).
Row 1 is A006498(n+2).
Row 2 is A189145(n+2).

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 12*n^2 - 6*n
k=4: a(n) = 36*n^2 - 36*n + 9
k=5: a(n) = 30*n^3 + 15*n^2 - 45*n + 15
k=6: a(n) = 25*n^4 + 50*n^3 - 25*n^2 - 50*n + 25
k=7: a(n) = 120*n^4 + 40*n^3 - 200*n^2 + 80*n

A207703 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 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, 15, 81, 102, 64, 10, 25, 225, 289, 216, 100, 12, 40, 625, 1071, 729, 390, 144, 14, 64, 1600, 3969, 3321, 1521, 636, 196, 16, 104, 4096, 13230, 15129, 8151, 2809, 966, 256, 18, 169, 10816, 44100, 61254, 43681, 17225, 4761, 1392

Offset: 1

R. H. Hardin, Feb 19 2012

Table starts
..2...4....6....9....15.....25......40.......64.......104.......169........273
..4..16...36...81...225....625....1600.....4096.....10816.....28561......74529
..6..36..102..289..1071...3969...13230....44100....153090....531441....1815939
..8..64..216..729..3321..15129...61254...248004...1050282...4447881...18510693
.10.100..390.1521..8151..43681..206910...980100...4863870..24137569..117661437
.12.144..636.2809.17225.105625..571350..3090564..17518470..99301225..552967815
.14.196..966.4761.32775.225625.1369900..8317456..52895444.336392281.2102483853
.16.256.1392.7569.57681.439569.2956980.19891600.140048460.986022801.6826106385

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

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

Column 2 is A016742.
Column 3 is A086113.
Column 4 is A207399.
Row 1 is A006498(n+2).
Row 2 is A189145(n+2).

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

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

A208039 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 102, 81, 13, 25, 225, 289, 279, 169, 19, 40, 625, 1071, 961, 741, 361, 28, 64, 1600, 3969, 4743, 3249, 1995, 784, 41, 104, 4096, 13230, 23409, 21147, 11025, 5404, 1681, 60, 169, 10816, 44100, 100215, 137641, 94605

Offset: 1

R. H. Hardin Feb 22 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
..9...81...279....961....4743....23409....100215.....429025.....1942075
.13..169...741...3249...21147...137641....766115....4264225....25232235
.19..361..1995..11025...94605...811801...5866411...42393121...327288437
.28..784..5404..37249..422477..4791721..44918280..421070400..4242161160
.41.1681.14555.126025.1889665.28334329.344414069.4186478209.55078752265

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

R. H. Hardin, Table of n, a(n) for n = 1..450

Column 1 is A000930(n+3)
Column 2 is A207170
Column 3 is A208023
Column 4 is A141583(n+3) for n>1
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207704

A208078 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 126, 81, 15, 26, 256, 510, 441, 225, 25, 42, 676, 1968, 2601, 1785, 625, 40, 68, 1764, 7722, 15129, 16065, 7225, 1600, 64, 110, 4624, 30114, 88209, 139605, 99225, 27880, 4096, 104, 178, 12100, 117708, 514089, 1228095

Offset: 1

R. H. Hardin, Feb 23 2012

			Table starts
..2....4......6......10.......16.........26..........42............68
..4...16.....36.....100......256........676........1764..........4624
..6...36....126.....510.....1968.......7722.......30114........117708
..9...81....441....2601....15129......88209......514089.......2996361
.15..225...1785...16065...139605....1228095....10751415......94313535
.25..625...7225...99225..1288225...17098225...224850025....2968615225
.40.1600..27880..572040.11204720..223058440..4412968520...87523832240
.64.4096.107584.3297856.97456384.2909955136.86610135616.2580469555456
...
Some solutions for n=4 k=3
..0..1..0....0..1..1....1..0..0....0..1..0....1..0..0....1..1..1....0..1..1
..1..0..1....1..1..0....1..1..0....0..1..1....0..1..1....1..0..1....1..1..0
..1..0..1....1..0..1....1..1..1....1..1..1....0..1..1....1..1..1....1..1..0
..0..1..1....0..1..1....1..1..1....1..0..0....1..1..1....0..1..1....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..716

Column 1 is A006498(n+2)
Column 2 is A189145(n+2)
Column 3 is A202399(n-2)
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A202954(n-2)

A208118 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 90, 81, 12, 25, 225, 225, 225, 144, 16, 40, 625, 825, 625, 420, 256, 20, 64, 1600, 3025, 3025, 1225, 784, 400, 25, 104, 4096, 9240, 14641, 7315, 2401, 1260, 625, 30, 169, 10816, 28224, 53361, 43681, 17689, 3969, 2025, 900

Offset: 1

R. H. Hardin Feb 23 2012

Table starts
..2...4....6....9....15.....25......40.......64.......104........169........273
..4..16...36...81...225....625....1600.....4096.....10816......28561......74529
..6..36...90..225...825...3025....9240....28224.....93912.....312481.....997815
..9..81..225..625..3025..14641...53361...194481....815409....3418801...13359025
.12.144..420.1225..7315..43681..175560...705600...3503640...17397241...76934095
.16.256..784.2401.17689.130321..577600..2560000..15054400...88529281..443060401
.20.400.1260.3969.34713.303601.1432600..6760000..45648200..308248249.1680152229
.25.625.2025.6561.68121.707281.3553225.17850625.138415225.1073283121.6371392041

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

R. H. Hardin, Table of n, a(n) for n = 1..1211

Column 1 is A002620(n+2)
Column 2 is A030179(n+2)
Column 3 is A207363
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207600

A209224 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 0 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 60, 81, 13, 25, 225, 100, 126, 169, 19, 40, 625, 240, 196, 234, 361, 28, 64, 1600, 576, 504, 324, 456, 784, 41, 104, 4096, 1296, 1296, 900, 576, 896, 1681, 60, 169, 10816, 2916, 3312, 2500, 1776, 1024, 1722, 3600, 88, 273

Offset: 1

R. H. Hardin Mar 06 2012

Table starts
..2....4....6....9...15....25....40.....64.....104.....169......273......441
..4...16...36...81..225...625..1600...4096...10816...28561....74529...194481
..6...36...60..100..240...576..1296...2916....6804...15876....36288....82944
..9...81..126..196..504..1296..3312...8464...21712...55696...142544...364816
.13..169..234..324..900..2500..6900..19044...52992..147456...407808..1127844
.19..361..456..576.1776..5476.15984..46656..143856..443556..1312020..3880900
.28..784..896.1024.3456.11664.38232.125316..423384.1430416..4755296.15808576
.41.1681.1722.1764.6300.22500.80400.287296.1037696.3748096.13540384.48916036

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

R. H. Hardin, Table of n, a(n) for n = 1..884

Column 1 is A000930(n+3)
Column 2 is A207170
Column 3 is A208496
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207694

A097083 Positive values of k such that there is exactly one permutation p of (1,2,3,...,k) such that i+p(i) is a Fibonacci number for 1<=i<=k.

Original entry on oeis.org

1, 2, 3, 5, 9, 15, 24, 39, 64, 104, 168, 272, 441, 714, 1155, 1869, 3025, 4895, 7920, 12815, 20736, 33552, 54288, 87840, 142129, 229970, 372099, 602069, 974169, 1576239, 2550408, 4126647, 6677056, 10803704, 17480760, 28284464, 45765225

Offset: 1

John W. Layman, Jul 23 2004

nonn

Numbers k such that A097082(k) = 1. If f is a Fibonacci number and k < f <= 2k, then a permutation for f-k-1 may be extended to a permutation for k, with p(i) = f-i for f-k < i <= k. This explains the sparseness of this sequence. - David Wasserman, Dec 19 2007

If the formula is correct, the bisections give A059840 and A064831. - David Wasserman, Dec 19 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,0,-1).

Cf. A073364, A000045.

a=b=c=d=0;Table[e=a+b+d+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
CoefficientList[Series[x/((x - 1)*(x^2 + 1)*(x^2 + x - 1)), {x,0,50}], x] (* G. C. Greubel, Mar 05 2017 *)
LinearRecurrence[{2,-1,1,0,-1},{1,2,3,5,9},50] (* Harvey P. Dale, Nov 09 2024 *)

x='x+O('x^50); Vec(x/((x - 1)*(x^2 + 1)*(x^2 + x - 1))) \\ G. C. Greubel, Mar 05 2017

It appears that {a(n)} satisfies a(1)=1, a(2)=2 and, for n>2, a(n) = F(n+2) - a(n-2) - 1, where {F(k)} is the sequence of Fibonacci numbers, i.e, that the sequence is the partial sums of A006498.
If the partial sum assumption is correct: a(n) = floor(phi^(n+3)/5), where phi=(1+sqrt(5))/2 = A001622, and a(n) = a(n-1) + a(n-2) + ( (n*(n+1)/2) mod 2). - Gary Detlefs, Mar 12 2011
From R. J. Mathar, Mar 13 2011: (Start)
If the partial sum assumption is correct: a(n)= +2*a(n-1) -a(n-2) +a(n-3) -a(n-5).
G.f.: x/( (x-1)*(x^2+1)*(x^2+x-1) ).
a(n) = A000032(n+3)/5 -(-1)^n*A112030(n)/10 - 1/2. (End)
Conjecture: a(n) = floor(F(n+3)/sqrt(5)), where F(n) = A000045(n) are Fibonacci numbers. - Vladimir Reshetnikov, Nov 05 2015

a(9) from Ray Chandler, Jul 29 2004
More terms from David Wasserman, Dec 19 2007
Terms > 90000 assuming the partial sums formula by Vladimir Joseph Stephan Orlovsky, Feb 26 2011

cp's OEIS Frontend

A105774 A "fractal" transform of the Fibonacci numbers: a(1)=1; then if F(n) < k <= F(n+1), a(k) = F(n+1) - a(k - F(n)) where F(n) = A000045(n).

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A207599 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 0 1 vertically.

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A207703 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 1 0 vertically.

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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.

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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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A208039 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 1 0 1 vertically.

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A208078 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 0 1 0 vertically.

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A208118 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 1 vertically.

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A209224 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 0 and 1 1 1 vertically.

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A097083 Positive values of k such that there is exactly one permutation p of (1,2,3,...,k) such that i+p(i) is a Fibonacci number for 1<=i<=k.

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