A138977 -id:A138977 - cp's OEIS frontend

A223504 T(n,k)=Petersen graph (3,1) coloring a rectangular array: number of nXk 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.

1, 3, 6, 9, 19, 36, 27, 115, 121, 216, 81, 631, 1519, 771, 1296, 243, 3539, 16323, 20115, 4913, 7776, 729, 19759, 182901, 426359, 266419, 31307, 46656, 2187, 110427, 2030665, 9685063, 11148439, 3528715, 199497, 279936, 6561, 617015, 22598167

Offset: 1

R. H. Hardin Mar 21 2013

Table starts
........1........3............9..............27.................81
........6.......19..........115.............631...............3539
.......36......121.........1519...........16323.............182901
......216......771........20115..........426359............9685063
.....1296.....4913.......266419........11148439..........515473927
.....7776....31307......3528715.......291545903........27465794119
....46656...199497.....46737819......7624417031......1463848507173
...279936..1271251....619042315....199391762123.....78024299447333
..1679616..8100769...8199214219...5214442630935...4158831849750231
.10077696.51620379.108598575915.136366781617267.221674060909378867

			Some solutions for n=3 k=4
..0..3..4..1....0..2..1..4....0..3..0..3....0..2..1..2....0..1..4..3
..0..3..4..3....5..2..5..4....4..1..0..1....1..2..0..2....0..1..0..3
..5..3..0..1....1..2..1..2....0..1..0..1....5..2..0..2....0..3..0..1

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 1 is A000400(n-1)
Column 2 is A138977
Row 1 is A000244(n-1)

Empirical for column k:
k=1: a(n) = 6*a(n-1)
k=2: a(n) = 7*a(n-1) -4*a(n-2)
k=3: a(n) = 15*a(n-1) -24*a(n-2) +10*a(n-3)
k=4: a(n) = 31*a(n-1) -127*a(n-2) -20*a(n-3) +705*a(n-4) -1027*a(n-5) +499*a(n-6) -60*a(n-7)
k=5: [order 21]
k=6: [order 53]
Empirical for row n:
n=1: a(n) = 3*a(n-1)
n=2: a(n) = 5*a(n-1) +4*a(n-2) -4*a(n-3) for n>4
n=3: a(n) = 12*a(n-1) -4*a(n-2) -73*a(n-3) +103*a(n-4) -23*a(n-5) -16*a(n-6) +4*a(n-7) for n>8
n=4: [order 21] for n>22
n=5: [order 60] for n>61

A214101 T(n,k)=Number of 0..2 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 1, 3, 3, 2, 9, 5, 19, 4, 27, 11, 30, 121, 8, 81, 21, 143, 180, 771, 16, 243, 43, 322, 2041, 1080, 4913, 32, 729, 85, 1179, 5068, 29540, 6480, 31307, 64, 2187, 171, 3110, 37441, 79968, 428383, 38880, 199497, 128, 6561, 341, 10183, 121588, 1241355, 1262128

Offset: 1

R. H. Hardin Jul 04 2012

Table starts
..1..1....3....5.....11......21.......43........85........171.........341
..3..2...19...30....143.....322.....1179......3110......10183.......28842
..9..4..121..180...2041....5068....37441....121588.....722009.....2720828
.27..8..771.1080..29540...79968..1241355...4807928...54733587...263068168
.81.16.4913.6480.428383.1262128.41634729.190532944.4254090231.25595530224

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

R. H. Hardin, Table of n, a(n) for n = 1..309

Column 3 is A138977
Column 4 is A052934
Row 1 is A001045
Row 2 is A094554(n+1)

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 2*a(n-1)
k=3: a(n) = 7*a(n-1) -4*a(n-2)
k=4: a(n) = 6*a(n-1)
k=5: a(n) = 19*a(n-1) -71*a(n-2) +86*a(n-3) -24*a(n-4)
k=6: a(n) = 18*a(n-1) -36*a(n-2) +16*a(n-3)
k=7: a(n) = 54*a(n-1) -820*a(n-2) +4906*a(n-3) -11803*a(n-4) +11888*a(n-5) -4672*a(n-6) +576*a(n-7)
Empirical for row n:
n=1: a(k)=a(k-1)+2*a(k-2)
n=2: a(k)=2*a(k-1)+5*a(k-2)-6*a(k-3)
n=3: a(k)=3*a(k-1)+15*a(k-2)-33*a(k-3)-22*a(k-4)+38*a(k-5)+8*a(k-6)-8*a(k-7)
n=4: (order 11)
n=5: (order 29)
n=6: (order 40)

A209517 T(n,k) = Number of (n+1) X (k+1) 0..2 arrays with the number of clockwise edge increases in every 2 X 2 subblock equal to the number of counterclockwise edge increases.

Original entry on oeis.org

57, 363, 363, 2313, 4995, 2313, 14739, 68937, 68937, 14739, 93921, 951777, 2071311, 951777, 93921, 598491, 13141335, 62340543, 62340543, 13141335, 598491, 3813753, 181445643, 1876947141, 4099186545, 1876947141, 181445643, 3813753, 24302307

Offset: 1

R. H. Hardin, Mar 09 2012

Table starts
.......57.........363...........2313.............14739.................93921
......363........4995..........68937............951777..............13141335
.....2313.......68937........2071311..........62340543............1876947141
....14739......951777.......62340543........4099186545..........269845393359
....93921....13141335.....1876947141......269845393359........38883961392081
...598491...181445643....56515446141....17769569448759......5607670578249789
..3813753..2505266673..1701725703867..1170258116647953....808953988260058047
.24302307.34590865185.51240520677831.77072446894479573.116711118063701004591

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

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is 3*A138977(n+1).
Column 2 is 3*A138978(n+1).
Column 3 is 3*A138979(n+1).

A138978 Number of 3 X n matrices containing a 1 in the top left entry, all entries are integer values and adjacent entries differ by at most 1.

Original entry on oeis.org

9, 121, 1665, 22979, 317259, 4380445, 60481881, 835088891, 11530288395, 159201677509, 2198138788809, 30350271502115, 419054058355851, 5785987905016141, 79888633386248025, 1103043049708026539, 15230001039404897259, 210284568423392013685, 2903453493049800669321

Offset: 1

Wayne VanWeerthuizen, Apr 05 2008

Horizontally or vertically adjacent entries can differ by at most 1. Diagonally adjacent entries thus differ by at most 2.

Alois P. Heinz, Table of n, a(n) for n = 1..800
Index entries for linear recurrences with constant coefficients, signature (16,-31,10).

Cf. A138977, A138979.

a:= n-> (Matrix([1,6,2]). Matrix([[3,12,2], [2,10,2], [1,6,3]])^(n-1) .Matrix([[1],[1],[1]]))[1,1]: seq(a(n), n=1..20); # Alois P. Heinz, Aug 28 2008

LinearRecurrence[{16, -31, 10}, {9, 121, 1665}, 25] (* Paolo Xausa, Mar 17 2024 *)

a(n) = b(n)+c(n)+d(n), where b(1)=1, c(1)=6, d(1)=2, with b(n+1)=3*b(n)+2*c(n)+1*d(n), c(n+1)=12*b(n)+10*c(n)+6*d(n), d(n+1)=2*b(n)+2*c(n)+3*d(n).

G.f.: -x*(8*x^2-23*x+9) / (10*x^3-31*x^2+16*x-1). - Colin Barker, Dec 03 2012

More terms from Alois P. Heinz, Aug 28 2008

A138979 Number of 4 X n matrices containing a 1 in the top left entry, all entries are integer values and adjacent entries differ by at most 1.

Original entry on oeis.org

27, 771, 22979, 690437, 20780181, 625649047, 18838482047, 567241901289, 17080173559277, 514300085627023, 15486061794514775, 466299978310573033, 14040733816061115637, 422779788989982722559, 12730299739840800975879, 383321378409770250813777

Offset: 1

Wayne VanWeerthuizen, Apr 05 2008

Horizontally or vertically adjacent entries can differ by at most 1. Diagonally adjacent entries thus differ by at most 2.

Alois P. Heinz, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (45,-528,2592,-5997,5689,812,-4760,1942,278,-112).

Cf. A138977, A138978.

a:= n-> (Matrix([2,4,4,2,4,2,2,4,2,1]). Matrix([[3,4,2,2,2,0,1,0,1,0], [2,5,3,2,3,1,1,3,2,1], [1,3,5,1,4,2,2,4,2,1], [2,4,2,3,4,2,2,4,2,1], [1,3,4,2,7,3,3,6,3,2], [0,2,4,2,6,4,2,6,2,2], [1,2,4,2,6,2,4,6,2,2], [0,3,4,2,6,3,3,7,3,2], [1,4,4,2,6,2,2,6,4,2], [0,4,4,2,8,4,4,8,4,3]])^(n-1) .Matrix([[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]]))[1,1]: seq(a(n), n=1..20); # Alois P. Heinz, Aug 28 2008

LinearRecurrence[{45, -528, 2592, -5997, 5689, 812, -4760, 1942, 278, -112}, {27, 771, 22979, 690437, 20780181, 625649047, 18838482047, 567241901289, 17080173559277, 514300085627023}, 20] (* Paolo Xausa, Mar 17 2024 *)

a(n)=b(n)+c(n)+d(n)+e(n)+f(n)+g(n)+h(n)+j(n)+k(n)+l(n), where
b(1)=2,c(1)=4,d(1)=4,e(1)=2,f(1)=4,g(1)=2,h(1)=2,j(1)=4,k(1)=2,l(1)=1
b(n+1)=3*b(n)+2*c(n)+1*d(n)+2*e(n)+1*f(n)+0*g(n)+1*h(n)+0*j(n)+1*k(n)+0*l(n)
c(n+1)=4*b(n)+5*c(n)+3*d(n)+4*e(n)+3*f(n)+2*g(n)+2*h(n)+3*j(n)+4*k(n)+4*l(n)
d(n+1)=2*b(n)+3*c(n)+5*d(n)+2*e(n)+4*f(n)+4*g(n)+4*h(n)+4*j(n)+4*k(n)+4*l(n)
e(n+1)=2*b(n)+2*c(n)+1*d(n)+3*e(n)+2*f(n)+2*g(n)+2*h(n)+2*j(n)+2*k(n)+2*l(n)
f(n+1)=2*b(n)+3*c(n)+4*d(n)+4*e(n)+7*f(n)+6*g(n)+6*h(n)+6*j(n)+6*k(n)+8*l(n)
g(n+1)=0*b(n)+1*c(n)+2*d(n)+2*e(n)+3*f(n)+4*g(n)+2*h(n)+3*j(n)+2*k(n)+4*l(n)
h(n+1)=1*b(n)+1*c(n)+2*d(n)+2*e(n)+3*f(n)+2*g(n)+4*h(n)+3*j(n)+2*k(n)+4*l(n)
j(n+1)=0*b(n)+3*c(n)+4*d(n)+4*e(n)+6*f(n)+6*g(n)+6*h(n)+7*j(n)+6*k(n)+8*l(n)
k(n+1)=1*b(n)+2*c(n)+2*d(n)+2*e(n)+3*f(n)+2*g(n)+2*h(n)+3*j(n)+4*k(n)+4*l(n)
l(n+1)=0*b(n)+1*c(n)+1*d(n)+1*e(n)+2*f(n)+2*g(n)+2*h(n)+2*j(n)+2*k(n)+3*l(n).
G.f.: -x*(-27 +116*x^9 -206*x^8 +5284*x^6 -2464*x^7 -154*x^5 +6514*x^3 -6915*x^4 -2540*x^2 +444*x) / (1 -45*x -1942*x^8 +528*x^2 -278*x^9 -2592*x^3 +112*x^10 +5997*x^4 -5689*x^5 -812*x^6 +4760*x^7). - Alois P. Heinz, Sep 02 2014

More terms from Alois P. Heinz, Aug 28 2008

A222169 T(n,k)=Number of nXk 0..4 arrays with entries increasing mod 5 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

Original entry on oeis.org

1, 3, 3, 9, 19, 9, 27, 121, 121, 27, 81, 771, 1665, 771, 81, 243, 4913, 22979, 22979, 4913, 243, 729, 31307, 317259, 690437, 317259, 31307, 729, 2187, 199497, 4380445, 20780181, 20780181, 4380445, 199497, 2187, 6561, 1271251, 60481881, 625649047

Offset: 1

R. H. Hardin Feb 10 2013

Table starts
......1..........3..............9.................27.....................81
......3.........19............121................771...................4913
......9........121...........1665..............22979.................317259
.....27........771..........22979.............690437...............20780181
.....81.......4913.........317259...........20780181.............1366395515
....243......31307........4380445..........625649047............89948464453
....729.....199497.......60481881........18838482047..........5923189816253
...2187....1271251......835088891.......567241901289........390086038882651
...6561....8100769....11530288395.....17080173559277......25690815631493191
..19683...51620379...159201677509....514300085627023....1691995329032459285
..59049..328939577..2198138788809..15486061794514775..111434983000652039093
.177147.2096095523.30350271502115.466299978310573033.7339124863989795685471

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

R. H. Hardin, Table of n, a(n) for n = 1..180

Diagonal is A068748
Column 1 is A000244(n-1)
Column 2 is A138977
Column 3 is A138978
Column 4 is A138979

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 7*a(n-1) -4*a(n-2)
k=3: a(n) = 16*a(n-1) -31*a(n-2) +10*a(n-3)
k=4: [order 10]
k=5: [order 25]
k=6: [order 70]

cp's OEIS Frontend

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A214101 T(n,k)=Number of 0..2 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..2 introduced in row major order.

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A209517 T(n,k) = Number of (n+1) X (k+1) 0..2 arrays with the number of clockwise edge increases in every 2 X 2 subblock equal to the number of counterclockwise edge increases.

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A138978 Number of 3 X n matrices containing a 1 in the top left entry, all entries are integer values and adjacent entries differ by at most 1.

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Maple

Mathematica

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A138979 Number of 4 X n matrices containing a 1 in the top left entry, all entries are integer values and adjacent entries differ by at most 1.

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Maple

Mathematica

Formula

Extensions

A222169 T(n,k)=Number of nXk 0..4 arrays with entries increasing mod 5 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

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