A189145 -id:A189145 - cp's OEIS frontend

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.

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

A264195 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having index change +-(.,.) 0,0 0,2 or 2,1.

Original entry on oeis.org

1, 4, 2, 16, 18, 4, 36, 180, 81, 8, 81, 864, 2025, 360, 16, 225, 4608, 20736, 19845, 1600, 32, 625, 29040, 262144, 395280, 194481, 6760, 64, 1600, 184525, 3748096, 10764800, 7535025, 1944810, 28561, 128, 4096, 1089000, 54479161, 324210304, 442050625

Offset: 1

R. H. Hardin, Nov 07 2015

Table starts
..1.....4.......16.........36...........81.............225.............625
..2....18......180........864.........4608...........29040..........184525
..4....81.....2025......20736.......262144.........3748096........54479161
..8...360....19845.....395280.....10764800.......324210304......9863496016
.16..1600...194481....7535025....442050625.....28044191296...1785793904896
.32..6760..1944810..150305220..19169627850...2585148634024.357076938417216
.64.28561.19448100.2998219536.831295356516.238302234835681

			Some solutions for n=3 k=4
..2..1.13..3..4...11..1.13..3..4...11..1.13.14..4....0.12..2.14..4
..5.17..7..6..9....5..6..9.19..7....5..6..9.19..7....7..6..5..8..9
.10..0.14.11.12...10..0.14..2.12...12..0.10..2..3...10.13..1.11..3
.15.16.19.18..8...15.18.17.16..8...15.16.17.18..8...17.18.15.16.19

R. H. Hardin, Table of n, a(n) for n = 1..83

Column 1 is A000079(n-1).

Row 1 is A189145(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: [order 14]
k=3: a(n) = 10*a(n-1) for n>5
k=4: [order 14] for n>16
k=5: [order 58]
Empirical for row n:
n=1: a(n) = 3*a(n-1) -3*a(n-2) +6*a(n-3) -6*a(n-5) +3*a(n-6) -3*a(n-7) +a(n-8)
n=2: [order 44]
n=3: [order 30]

A197424 Number of subsets of {1, 2, ..., 4*n + 2} which do not contain two numbers whose difference is 4.

Original entry on oeis.org

4, 36, 225, 1600, 10816, 74529, 509796, 3496900, 23961025, 164249856, 1125736704, 7716041281, 52886200900, 362488284900, 2484529385121, 17029223715904, 116720020119616, 800010960336225, 5483356589096100, 37583485459535236, 257601040852192129

Offset: 0

John W. Layman, Oct 14 2011

This sequence is an instance of a general result given in Math. Mag. Problem 1854 (see Links).
From Feryal Alayont, May 20 2023: (Start)
a(n) is the number of edge covers of a caterpillar graph with spine P_(4n+5), one pendant attached at vertex n+2 counting from the left end of the spine, a second one at 2n+3 and a third at 3n+4. The caterpillar graph for n=1 is as follows:
        *      *      *
        |      |      |
  *--*--*--v1--*--v2--*--*--*
Each pendant edge must be included in an edge cover leaving only the middle six edges flexible. Every vertex except v1 and v2 is incident with at least one of the pendant edges. Therefore, if we label the middle six edges in the spine with numbers 3, 1, 5, 2, 6, 4 (starting from the left), the edges have to be chosen so that both 1,5 and 2,6 cannot be missing. This corresponds to choosing subsets of {1, 2, ..., 6} which do not contain two numbers whose difference is 4. (End)

F. Alayont and E. Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs; submitted.

Colin Barker, Table of n, a(n) for n = 0..1000
Mathramz Problem Solving Group, Solution for Problem 1854 in Mathematics Magazine Vol. 83, No.4, October 2010, February 28, 2011
Marian Tetiva, Problem 1854, Mathematics Magazine, 84 (2011) 300.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000045, A001654, A060635, A066258.

Table[(1/25) (LucasL[2 (2 n + 5)] - 2 (-1)^n LucasL[2 n + 5] - 1), {n, 0, 20}] (* Michael De Vlieger, Mar 27 2016 *)

Vec((4+16*x-15*x^2-5*x^3+x^4) / ((1-x)*(1-7*x+x^2)*(1+3*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 26 2016

a(n) = (fibonacci(n+2)*fibonacci(n+3))^2; \\ Altug Alkan, Mar 26 2016

a(n) = F(n+2)^2*F(n+3)^2 = A001654(n+2)^2, where F(n) denotes the n-th Fibonacci number A000045(n).
G.f.: ( -4-16*x+15*x^2+5*x^3-x^4 ) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - R. J. Mathar, Oct 15 2011
Empirical: a(n) = A189145(2n+3). - R. J. Mathar, Oct 15 2011
For L=Lucas, a(n) = (1/25)*(L(2*(2*n+5)) - 2*(-1)^n*L(2*n+5) - 1), an instance of (F(n+p)*F(n+q))^2 = (1/25)*(L(2*(2*n+p+q)) - 2*(-1)^(n+q)*L(p-q)*L(2*n+p+q) + L(2*(p-q)) + 4*(-1)^(p-q)) which follows from squaring a specialization of identity 17b in the Vajda reference at A000045, F(n+p)*F(n+q) = (1/5)*(L(2*n+p+q) - (-1)*(n+q)*L(p-q)), then applying Vajda 17c, L(n)^2 = L(2*n) + 2*(-1)^n, to the expansion. - Ehren Metcalfe, Mar 26 2016
a(n) = A060635(n+2)/2. - Alois P. Heinz, Jul 03 2025

A201222 Number of ways to place k non-attacking knights on a 2 X n horizontal cylinder, summed over all k>=0.

Original entry on oeis.org

3, 9, 18, 81, 123, 324, 843, 2401, 5778, 15129, 39603, 104976, 271443, 710649, 1860498, 4879681, 12752043, 33385284, 87403803, 228886641, 599074578, 1568397609, 4106118243, 10750371856, 28143753123, 73681302249, 192900153618, 505022001201, 1322157322203

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces

Cf. A189145, A001333, A051927, A006506, A067966.

Table[If[Mod[n,4]==0,LucasL[n/2]^4,LucasL[2n]+1+(-1)^n],{n,1,50}]

A321251 a(n) is the number of ways to place non-attacking knights on a 3 X n chessboard.

Original entry on oeis.org

1, 8, 36, 94, 278, 1062, 3650, 11856, 39444, 135704, 456980, 1534668, 5166204, 17480600, 58888528, 198548648, 669291696, 2258436248, 7613387344, 25676313144, 86575342536, 291991130840, 984557555352, 3320284572360, 11196209499736, 37757232570616

Offset: 0

Dimitrios Noulas, Nov 01 2018

For n = 3, a(3) = 94 is the same as A141243(3). In both cases these are 3 X 3 chessboards.

Dimitrios Noulas, Table of n, a(n) for n = 0..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 297-301.

Cf. A141243, A189145.

CoefficientList[Series[-(36 x^15 - 72 x^14 - 60 x^13 + 72 x^12 - 120 x^11 + 250 x^10 + 270 x^9 - 256 x^8 - 30 x^7 - 78 x^6 - 98 x^5 + 92 x^4 + 36 x^3 - 8 x^2 - 5 x - 1)/(36 x^16 - 108 x^15 + 48 x^14 + 24 x^13 - 144 x^12 + 376 x^11 - 70 x^10 - 174 x^9 + 108 x^8 - 168 x^7 + 26 x^6 + 78 x^5 - 24 x^4 + 10 x^3 - 4 x^2 - 3 x + 1), {x, 0, 25}], x] (* Michael De Vlieger, Nov 05 2018 *)

G(x)=-(36*x^15 - 72*x^14 - 60*x^13 + 72*x^12 - 120*x^11 + 250*x^10 + 270*x^9 - 256*x^8 - 30*x^7 - 78*x^6 - 98*x^5 + 92*x^4 + 36*x^3 - 8*x^2 - 5*x - 1)/(36*x^16 - 108*x^15 + 48*x^14 + 24*x^13 - 144*x^12 + 376*x^11 - 70*x^10 - 174*x^9 + 108*x^8 - 168*x^7 + 26*x^6 + 78*x^5 - 24*x^4 + 10*x^3 - 4*x^2 - 3*x + 1)
G.series(x,1001)

G.f.: -(36*x^15 - 72*x^14 - 60*x^13 + 72*x^12 - 120*x^11 + 250*x^10 + 270*x^9 - 256*x^8 - 30*x^7 - 78*x^6 - 98*x^5 + 92*x^4 + 36*x^3 - 8*x^2 - 5*x - 1)/(36*x^16 - 108*x^15 + 48*x^14 + 24*x^13 - 144*x^12 + 376*x^11 - 70*x^10 - 174*x^9 + 108*x^8 - 168*x^7 + 26*x^6 + 78*x^5 - 24*x^4 + 10*x^3 - 4*x^2 - 3*x + 1).

cp's OEIS Frontend

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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A264195 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having index change +-(.,.) 0,0 0,2 or 2,1.

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Author

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A197424 Number of subsets of {1, 2, ..., 4*n + 2} which do not contain two numbers whose difference is 4.

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Mathematica

PARI

PARI

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A201222 Number of ways to place k non-attacking knights on a 2 X n horizontal cylinder, summed over all k>=0.

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Mathematica

A321251 a(n) is the number of ways to place non-attacking knights on a 3 X n chessboard.

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Mathematica

Sage

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