A199560 -id:A199560 - cp's OEIS frontend

A208392 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

1, 2, 2, 5, 14, 5, 14, 117, 122, 14, 41, 1017, 3042, 1094, 41, 122, 8838, 76806, 79092, 9842, 122, 365, 76806, 1937736, 5800644, 2056392, 88574, 365, 1094, 667476, 48890520, 424785708, 438083928, 53466192, 797162, 1094, 3281, 5800644

Offset: 1

R. H. Hardin Feb 25 2012

Table starts
....1.......2...........5..............14.................41
....2......14.........117............1017...............8838
....5.....122........3042...........76806............1937736
...14....1094.......79092.........5800644..........424785708
...41....9842.....2056392.......438083928........93120350760
..122...88574....53466192.....33085555344.....20413586117376
..365..797162..1390120992...2498731184736...4475009970818208
.1094.7174454.36143145792.188712490047552.980999326809336384

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

R. H. Hardin, Table of n, a(n) for n = 1..301

Column 1 is A007051(n-1)
Column 2 is A199560(n-1)
Row 1 is A007051(n-1)

A233082 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally or antidiagonally, top left element zero, and 1 appearing before 2 in row major order.

Original entry on oeis.org

1, 2, 3, 5, 14, 10, 14, 95, 122, 36, 41, 662, 1985, 1094, 136, 122, 4631, 32414, 41675, 9842, 528, 365, 32414, 529862, 1588262, 875165, 88574, 2080, 1094, 226895, 8662343, 60632429, 77824814, 18378455, 797162, 8256, 3281, 1588262, 141615905

Offset: 1

R. H. Hardin, Dec 03 2013

Table starts
......1.........2.............5................14....................41
......3........14............95...............662..................4631
.....10.......122..........1985.............32414................529862
.....36......1094.........41675...........1588262..............60632429
....136......9842........875165..........77824814............6938214854
....528.....88574......18378455........3813415862..........793945203881
...2080....797162.....385947545......186857377214........90851753687090
...8256...7174454....8104898435.....9156011483462.....10396235291448605
..32896..64570082..170202867125...448644562689614...1189649113515482414
.131328.581130734.3574260209615.21983583571791062.136132453105625552657

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

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

Column 1 is A007582(n-1)
Column 2 is A199560(n-1)
Row 1 is A007051(n-1)

Empirical for column k:
k=1: a(n) = 6*a(n-1) -8*a(n-2)
k=2: a(n) = 10*a(n-1) -9*a(n-2)
k=3: a(n) = 22*a(n-1) -21*a(n-2)
k=4: a(n) = 50*a(n-1) -49*a(n-2)
k=5: a(n) = 118*a(n-1) -411*a(n-2) +294*a(n-3)
k=6: a(n) = 283*a(n-1) -4251*a(n-2) +13573*a(n-3) -9604*a(n-4)
k=7: [order 6]
Empirical for row n:
n=1: a(n) = 4*a(n-1) -3*a(n-2)
n=2: a(n) = 8*a(n-1) -7*a(n-2) for n>3
n=3: a(n) = 19*a(n-1) -45*a(n-2) +27*a(n-3) for n>5
n=4: a(n) = 49*a(n-1) -450*a(n-2) +1466*a(n-3) -1853*a(n-4) +789*a(n-5) for n>8
n=5: [order 10] for n>14
n=6: [order 21] for n>26
n=7: [order 52] for n>58

A199561 a(n) = 3*9^n + 1.

Original entry on oeis.org

4, 28, 244, 2188, 19684, 177148, 1594324, 14348908, 129140164, 1162261468, 10460353204, 94143178828, 847288609444, 7625597484988, 68630377364884, 617673396283948, 5559060566555524, 50031545098999708, 450283905890997364, 4052555153018976268, 36472996377170786404

Offset: 0

Vincenzo Librandi, Nov 08 2011

An Engel expansion of 3 to the base 9 as defined in A181565, with the associated series expansion 3 = 9/4 + 9^2/(4*28) + 9^3/(4*28*244) + 9^4/(4*28*244*2188) + .... Cf. A087289 and A207262. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A066443, A087289, A181565, A199560, A207262.

Magma
```
[3*9^n+1: n in [0..30]];
```

3*9^Range[0,20]+1 (* or *) LinearRecurrence[{10,-9},{4,28},20] (* Harvey P. Dale, Jul 30 2019 *)

a(n) = 4*A066443(n).
a(n) = 9*a(n-1) - 8.
a(n) = 10*a(n-1) - 9*a(n-2).
G.f.: 4*(1-3*x)/((1-x)*(1-9*x)).
From Elmo R. Oliveira, Sep 13 2024: (Start)
E.g.f.: exp(x)*(3*exp(8*x) + 1).
a(n) = 2*A199560(n). (End)

cp's OEIS Frontend

A208392 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A233082 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally or antidiagonally, top left element zero, and 1 appearing before 2 in row major order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A199561 a(n) = 3*9^n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula