A159739 -id:A159739 - cp's OEIS frontend

A269289 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling three no more than once.

4, 16, 16, 60, 216, 64, 216, 2124, 2592, 256, 756, 19188, 62748, 29160, 1024, 2592, 164556, 1363572, 1698732, 314928, 4096, 8748, 1363572, 27788292, 87559668, 43674876, 3306744, 16384, 29160, 11026764, 544118148, 4204943820, 5306911092

Offset: 1

R. H. Hardin, Feb 21 2016

Table starts
......4.........16.............60...............216...................756
.....16........216...........2124.............19188................164556
.....64.......2592..........62748...........1363572..............27788292
....256......29160........1698732..........87559668............4204943820
...1024.....314928.......43674876........5306911092..........598478857956
...4096....3306744.....1085203980......309846524148........81907569617580
..16384...34012224....26317946844....17623065834612.....10908770041709316
..65536..344373768...626778812268...983118947312628...1424067311317705740
.262144.3443737680.14718495557052.54032675767734132.183070424003703987492

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

R. H. Hardin, Table of n, a(n) for n = 1..241

Column 1 is A000302.
Column 2 is A159739(n+1).
Row 1 is A120926(n+1).

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 18*a(n-1) -81*a(n-2)
k=3: a(n) = 42*a(n-1) -441*a(n-2)
k=4: a(n) = 98*a(n-1) -2401*a(n-2) for n>3
k=5: a(n) = 234*a(n-1) -14277*a(n-2) +68796*a(n-3) -86436*a(n-4)
k=6: [order 6] for n>7
k=7: [order 10] for n>11
Empirical for row n:
n=1: a(n) = 6*a(n-1) -9*a(n-2)
n=2: a(n) = 14*a(n-1) -49*a(n-2) for n>4
n=3: a(n) = 36*a(n-1) -378*a(n-2) +972*a(n-3) -729*a(n-4) for n>7
n=4: [order 8] for n>12
n=5: [order 18] for n>23
n=6: [order 40] for n>46

A159721 Number of permutations of 3 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

6, 36, 192, 960, 4608, 21504, 98304, 442368, 1966080, 8650752, 37748736, 163577856, 704643072, 3019898880, 12884901888, 54760833024, 231928233984, 979252543488, 4123168604160, 17317308137472, 72567767433216, 303465209266176, 1266637395197952, 5277655813324800

Offset: 2

R. H. Hardin, Apr 20 2009

R. H. Hardin, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A159715, A159727, A159733, A159736, A159738, A159739, A159740.

[3*n*4^(n-2): n in [2..30]]; // G. C. Greubel, Jun 01 2018

LinearRecurrence[{8,-16}, {6,36}, 30] (* or *) Table[3*n*4^(n-2), {n, 2, 30}] (* G. C. Greubel, Jun 01 2018 *)

for(n=2,30, print1(3*n*4^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

a(n) = (copies*n)*(copies+1)^(n-2), here: copies = 3.
From Colin Barker, Mar 23 2018: (Start)
G.f.: 6*x^2*(1 - 2*x) / (1 - 4*x)^2.
a(n) = 8*a(n-1) - 16*a(n-2) for n>3. (End)
E.g.f.: 3*x*exp(4*x)/4. - G. C. Greubel, Jun 01 2018
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (16/3)*log(4/3) - 3/2.
Sum_{n>=2} (-1)^n/a(n) = (16/3)*log(5/4) - 7/6. (End)

A159727 Number of permutations of 4 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

8, 60, 400, 2500, 15000, 87500, 500000, 2812500, 15625000, 85937500, 468750000, 2539062500, 13671875000, 73242187500, 390625000000, 2075195312500, 10986328125000, 57983398437500, 305175781250000, 1602172851562500

Offset: 2

R. H. Hardin, Apr 20 2009

R. H. Hardin, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A159715, A159721, A159733, A159736, A159738, A159739, A159740.

[ 4*n*5^(n-2) : n in [2..30]]; // G. C. Greubel, Jun 01 2018

LinearRecurrence[{10,-25}, {8,60}, 30] (* or *) Table[4*n*5^(n-2), {n,2,30}] (* G. C. Greubel, Jun 01 2018 *)

for(n=2, 30, print1(4*n*5^(n-2) , ", ")) \\ G. C. Greubel, Jun 01 2018

a(n) = (4*n)*(4+1)^(n-2).
From Colin Barker, Mar 23 2018: (Start)
G.f.: 4*x^2*(2 - 5*x) / (1 - 5*x)^2.
a(n) = 10*a(n-1) - 25*a(n-2) for n>3. (End)
E.g.f.: 4*x*exp(5*x)/5. - G. C. Greubel, Jun 01 2018
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (25/4)*log(5/4) - 5/4.
Sum_{n>=2} (-1)^n/a(n) = 5/4 - (25/4)*log(6/5). (End)

A159733 Number of permutations of 5 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

10, 90, 720, 5400, 38880, 272160, 1866240, 12597120, 83980800, 554273280, 3627970560, 23581808640, 152374763520, 979552051200, 6269133127680, 39965723688960, 253899891671040, 1608032647249920, 10155995666841600

Offset: 2

R. H. Hardin, Apr 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A159715, A159721, A159727, A159736, A159738, A159739, A159740.

[5*n*6^(n-2): n in [2..30]]; // G. C. Greubel, Jun 01 2018

LinearRecurrence[{12,-36}, {10,90}, 30] (* or *) Table[5*n*6^(n-2), {n,2,30}] (* G. C. Greubel, Jun 01 2018 *)

for(n=2,30, print1(5*n*6^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

a(n) = (copies*n)*(copies+1)^(n-2) where copies=5.
From Colin Barker, Mar 24 2018: (Start)
G.f.: 10*x^2*(1 - 3*x) / (1 - 6*x)^2.
a(n) = 12*a(n-1) - 36*a(n-2) for n>3. (End)
From G. C. Greubel, Jun 01 2018: (Start)
a(n) = 5*n*6^(n-2).
E.g.f.: 5*x*exp(6*x)/6. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (36/5)*log(6/5) - 6/5.
Sum_{n>=2} (-1)^n/a(n) = 6/5 - (36/5)*log(7/6). (End)

A159715 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

4, 18, 72, 270, 972, 3402, 11664, 39366, 131220, 433026, 1417176, 4605822, 14880348, 47829690, 153055008, 487862838, 1549681956, 4907326194, 15496819560, 48814981614, 153418513644, 481176247338, 1506290861232, 4707158941350

Offset: 2

R. H. Hardin, Apr 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A027261, A159721, A159727, A159733, A159736, A159738, A159739, A159740.

[2*n*3^(n-2): n in [2..30]]; // G. C. Greubel, Jun 01 2018

LinearRecurrence[{6,-9}, {}, 30] (* or *) Table[2*n*3^(n-2), {n, 2, 30}] (* G. C. Greubel, Jun 01 2018 *)

for(n=2, 30, print1(2*n*3^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

a(n) = (copies*n)*(copies+1)^(n-2), here: copies = 2.
Apparently a(n) = A027261(n-1), n > 2. - R. J. Mathar, Apr 21 2009
Conjectures from Colin Barker, Mar 23 2018: (Start)
G.f.: 2*x^2*(2 - 3*x) / (1 - 3*x)^2.
a(n) = 2*3^(n-2)*n for n>1.
a(n) = 6*a(n-1) - 9*a(n-2) for n>3. (End)
E.g.f.: 2*x*exp(3*x)/3. - G. C. Greubel, Jun 01 2018
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (9/2)*log(3/2) - 3/2.
Sum_{n>=2} (-1)^n/a(n) = 3/2 - (9/2)*log(4/3). (End)

A159736 Number of permutations of 6 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

12, 126, 1176, 10290, 86436, 705894, 5647152, 44471322, 345888060, 2663338062, 20338217928, 154231485954, 1162668124884, 8720010936630, 65109414993504, 484251274014186, 3589156501516908, 26519878594541598, 195409631749253880

Offset: 2

R. H. Hardin, Apr 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A159715, A159721, A159727, A159733, A159738, A159739, A159740.

I:=[12, 126]; [n le 2 select I[n] else 14*Self(n-1) - 49*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 01 2018

LinearRecurrence[{14,-49}, {12, 126}, 30] (* or *) Table[6*n*7^(n-2), {n, 2, 30}] (* G. C. Greubel, Jun 01 2018 *)

for(n=2, 30, print1(6*n*7^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

a(n) = (copies*n)*(copies+1)^(n-2).
From G. C. Greubel, Jun 01 2018: (Start)
a(n) = 6*n*7^(n-2).
a(n) = 14*a(n-1) - 49*a(n-2).
G.f.: x^2*(12-42*x)/(1-14*x+49*x^2).
E.g.f.: 6*x*exp(7*x)/7. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (49/6)*log(7/6) - 7/6.
Sum_{n>=2} (-1)^n/a(n) = 7/6 - (49/6)*log(8/7). (End)

A159738 Number of permutations of 7 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

14, 168, 1792, 17920, 172032, 1605632, 14680064, 132120576, 1174405120, 10334765056, 90194313216, 781684047872, 6734508720128, 57724360458240, 492581209243648, 4186940278571008, 35465847065542656, 299489375220137984

Offset: 2

R. H. Hardin, Apr 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A159715, A159721, A159727, A159733, A159736, A159739, A159740.

I:=[14, 168]; [n le 2 select I[n] else 16*Self(n-1) - 64*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 01 2018

LinearRecurrence[{16,-64}, {14,168}, 30] (* or *) Table[7*n*8^(n-2), {n,2,30}] (* G. C. Greubel, Jun 01 2018 *)

for(n=2,30, print1(7*n*8^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

a(n) = (copies*n)*(copies+1)^(n-2).
From G. C. Greubel, Jun 01 2018: (Start)
a(n) = 7*n*8^(n-2).
a(n) = 16*a(n-1) - 64*a(n-2).
G.f.: x^2*(14-56*x)/(1-16*x+64*x^2).
E.g.f.: 7*x*exp(8*x)/8. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (64/7)*log(8/7) - 8/7.
Sum_{n>=2} (-1)^n/a(n) = 8/7 - (64/7)*log(9/8). (End)

A159740 Number of permutations of 9 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

18, 270, 3600, 45000, 540000, 6300000, 72000000, 810000000, 9000000000, 99000000000, 1080000000000, 11700000000000, 126000000000000, 1350000000000000, 14400000000000000, 153000000000000000

Offset: 2

R. H. Hardin, Apr 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A159715, A159721, A159727, A159733, A159736, A159738, A159739.

I:=[18, 270]; [n le 2 select I[n] else 20*Self(n-1) - 100*Self(n-2): n in [1..10]]; // G. C. Greubel, Jun 01 2018

LinearRecurrence[{20,-100}, {18,270}, 30] (* or *) Table[9*n*10^(n-2), {n,2,30}] (* G. C. Greubel, Jun 01 2018 *)

m=30; v=concat([18, 270], vector(m-2)); for(n=3, m, v[n]=20*v[n-1] -100*v[n-2]); v \\ G. C. Greubel, Jun 01 2018

a(n) = (copies*n)*(copies+1)^(n-2).
From G. C. Greubel, Jun 01 2018: (Start)
a(n) = 9*n*10^(n-2).
G.f.: x^2*(18-90*x)/(1-20*x+100*x^2).
E.g.f.: 9*x*exp(10*x)/10. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (100/9)*log(10/9) - 10/9.
Sum_{n>=2} (-1)^n/a(n) = 10/9 - (100/9)*log(11/10). (End)

cp's OEIS Frontend

A269289 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling three no more than once.

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A159721 Number of permutations of 3 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

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A159727 Number of permutations of 4 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

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A159733 Number of permutations of 5 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

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A159715 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

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A159736 Number of permutations of 6 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

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A159738 Number of permutations of 7 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

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A159740 Number of permutations of 9 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

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