A239333 -id:A239333 - cp's OEIS frontend

A239405 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it, modulo 4.

2, 4, 5, 8, 23, 12, 16, 105, 129, 28, 32, 478, 1337, 698, 66, 64, 2165, 13977, 16449, 3805, 156, 128, 9811, 144762, 394509, 205969, 20818, 368, 256, 44399, 1504399, 9340070, 11334364, 2590704, 113774, 868, 512, 201006, 15591122, 222457036, 614216271

Offset: 1

R. H. Hardin, Mar 17 2014

Table starts
....2.......4..........8............16................32...................64
....5......23........105...........478..............2165.................9811
...12.....129.......1337.........13977............144762..............1504399
...28.....698......16449........394509...........9340070............222457036
...66....3805.....205969......11334364.........614216271..........33522573911
..156...20818....2590704.....326882784.......40566958960........5072216819342
..368..113774...32507376....9402398952.....2671846338458......765206237815409
..868..621754..408000104..270516990628...176044655676921...115478426410506313
.2048.3399032.5124790809.7789048968124.11610439554314901.17442858177037612521

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

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

Column 1 is A239333

Row 1 is A000079

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-3)
k=2: [order 16]
k=3: [order 64]
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 3*a(n-1) +12*a(n-2) -18*a(n-3) -29*a(n-4) +19*a(n-5) +38*a(n-6) +8*a(n-7)
n=3: [order 25]
n=4: [order 91]

A240000 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

2, 3, 5, 4, 13, 12, 5, 25, 61, 28, 6, 42, 190, 256, 66, 7, 65, 526, 1372, 1117, 156, 8, 95, 1262, 6527, 10405, 5012, 368, 9, 133, 2766, 27415, 86360, 83029, 22592, 868, 10, 180, 5647, 104291, 635873, 1225281, 685898, 102336, 2048, 11, 237, 10878, 363859

Offset: 1

R. H. Hardin, Mar 30 2014

Table starts
....2.......3.........4...........5............6............7............8
....5......13........25..........42...........65...........95..........133
...12......61.......190.........526.........1262.........2766.........5647
...28.....256......1372........6527........27415.......104291.......363859
...66....1117.....10405.......86360.......635873......4267171.....26152051
..156....5012.....83029.....1225281.....15981219....191691132...2090236137
..368...22592....685898....18392485....429788876...9314138750.182333502325
..868..102336...5825700...290513038..12392346376.491124025940
.2048..465662..50417154..4767970186.378942837634
.4832.2123857.441675344.80410934960

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

R. H. Hardin, Table of n, a(n) for n = 1..98

Column 1 is A239333

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-3)
k=2: [order 26]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = (1/6)*n^3 + 1*n^2 + (23/6)*n
n=3: [polynomial of degree 8] for n>6
n=4: [polynomial of degree 19] for n>20
n=5: [polynomial of degree 44] for n>52

A240460 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

2, 4, 5, 8, 21, 12, 16, 89, 102, 28, 32, 375, 874, 476, 66, 64, 1583, 7589, 8187, 2200, 156, 128, 6685, 65723, 143237, 75167, 10123, 368, 256, 28241, 568370, 2501883, 2632690, 682018, 46471, 868, 512, 119319, 4916340, 43654661, 92193017, 47636104, 6147372

Offset: 1

R. H. Hardin, Apr 05 2014

Table starts
....2.......4..........8...........16............32.............64
....5......21.........89..........375..........1583...........6685
...12.....102........874.........7589.........65723.........568370
...28.....476.......8187.......143237.......2501883.......43654661
...66....2200......75167......2632690......92193017.....3228706651
..156...10123.....682018.....47636104....3337483054...234122685500
..368...46471....6147372....854671234..119643524281.16795323356869
..868..213000...55212526..15259447330.4265146133201
.2048..975380..494809053.271656519692
.4832.4464474.4428808128

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

R. H. Hardin, Table of n, a(n) for n = 1..84

Column 1 is A239333

Row 1 is A000079

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-3)
k=2: [order 26]
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 6*a(n-1) -7*a(n-2) -4*a(n-3) +8*a(n-4)
n=3: [order 23]

A255115 Number of n-length words on {0,1,2} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 2, 5, 12, 28, 66, 156, 368, 868, 2048, 4832, 11400, 26896, 63456, 149712, 353216, 833344, 1966112, 4638656, 10944000, 25820224, 60917760, 143723520, 339087488, 800010496, 1887468032, 4453111040, 10506243072, 24787422208, 58481066496, 137974619136

Offset: 0

Milan Janjic, Feb 14 2015

Apparently a(n) = A239333(n).

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.
Index entries for linear recurrences with constant coefficients, signature (2,0,2).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 2,  a[2]== 5, a[n] == 2 a[n - 1] + 2 a[n - 3]}, a[n], {n, 0, 29}]

Vec(-(x^2+1)/(2*x^3+2*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 2*a(n+2) + 2*a(n) with n>1, a(0) = 1, a(1) = 2, a(2)=5.
G.f.: -(x^2+1) / (2*x^3+2*x-1). - Colin Barker, Feb 15 2015
a(n) = A052912(n)+A052912(n-2). - R. J. Mathar, Jun 18 2015

A255116 Number of n-length words on {0,1,2,3} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 3, 10, 33, 108, 354, 1161, 3807, 12483, 40932, 134217, 440100, 1443096, 4731939, 15516117, 50877639, 166828734, 547034553, 1793736576, 5881695930, 19286191449, 63239784075, 207364440015, 679951894392, 2229575035401, 7310818426248, 23972310961920

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.
Index entries for linear recurrences with constant coefficients, signature (3,0,3).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 3,  a[2]== 10, a[n] == 3 a[n - 1] + 3 a[n - 3]}, a[n], {n, 0, 25}]
LinearRecurrence[{3,0,3},{1,3,10},30] (* Harvey P. Dale, Feb 20 2023 *)

Vec(-(x^2+1)/(3*x^3+3*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 3*a(n+2) + 3*a(n) with n>1, a(0) = 1, a(1) = 3, a(2) = 10.
G.f.: -(x^2+1) / (3*x^3+3*x-1). - Colin Barker, Feb 15 2015
a(n) = A089978(n) + A089978(n-2). - R. J. Mathar, Aug 04 2019

A255117 Number of n-length words on {0,1,2,3,4} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 4, 17, 72, 304, 1284, 5424, 22912, 96784, 408832, 1726976, 7295040, 30815488, 130169856, 549859584, 2322700288, 9811480576, 41445360640, 175072243712, 739534897152, 3123921031168, 13195973099520, 55742031986688, 235463812071424, 994639140683776

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.
Index entries for linear recurrences with constant coefficients, signature (4,0,4).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 4,  a[2]== 17, a[n] == 4 a[n - 1] + 4 a[n - 3]}, a[n], {n, 0, 25}]

Vec(-(x^2+1)/(4*x^3+4*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 4*a(n+2) + 4*a(n) with n>1, a(0) = 1, a(1) = 4, a(2) = 17.
G.f.: -(x^2+1) / (4*x^3+4*x-1). - Colin Barker, Feb 15 2015
a(n) = A089979(n) + A089979(n-2). - R. J. Mathar, Aug 04 2019

A255118 Number of n-length words on {0,1,2,3,4,5} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 5, 26, 135, 700, 3630, 18825, 97625, 506275, 2625500, 13615625, 70609500, 366175000, 1898953125, 9847813125, 51069940625, 264844468750, 1373461409375, 7122656750000, 36937506093750, 191554837515625, 993387471328125, 5151624887109375, 26715898623125000

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10
Index entries for linear recurrences with constant coefficients, signature (5,0,5).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 5,  a[2]== 26, a[n] == 5 a[n - 1] + 5 a[n - 3]}, a[n], {n, 0, 20}]

Vec(-(x^2+1)/(5*x^3+5*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 5*a(n+2) + 5*a(n) with n>1, a(0) = 1, a(1) = 5, a(2) = 26.

G.f.: -(x^2+1) / (5*x^3+5*x-1). - Colin Barker, Feb 15 2015

A255119 Number of n-length words on {0,1,2,3,4,5,6} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 6, 37, 228, 1404, 8646, 53244, 327888, 2019204, 12434688, 76575456, 471567960, 2904015888, 17883548064, 110130696144, 678208272192, 4176550921536, 25720089706080, 158389787869632, 975398032747008, 6006708734718528, 36990591135528960

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3. example 10.
Index entries for linear recurrences with constant coefficients, signature (6,0,6).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 6,  a[2]== 37, a[n] == 6 a[n - 1] + 6 a[n - 3]}, a[n], {n, 0, 20}]
LinearRecurrence[{6,0,6},{1,6,37},30] (* Harvey P. Dale, Nov 06 2017 *)

Vec(-(x^2+1)/(6*x^3+6*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 6*a(n+2) + 6*a(n) with n>1, a(0) = 1, a(1) = 6, a(2) = 37.

G.f.: -(x^2+1) / (6*x^3+6*x-1). - Colin Barker, Feb 15 2015

cp's OEIS Frontend

A239405 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it, modulo 4.

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A240000 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4.

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A240460 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4.

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A255115 Number of n-length words on {0,1,2} in which 0 appears only in runs of length 2.

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A255116 Number of n-length words on {0,1,2,3} in which 0 appears only in runs of length 2.

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A255117 Number of n-length words on {0,1,2,3,4} in which 0 appears only in runs of length 2.

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A255118 Number of n-length words on {0,1,2,3,4,5} in which 0 appears only in runs of length 2.

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Mathematica

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A255119 Number of n-length words on {0,1,2,3,4,5,6} in which 0 appears only in runs of length 2.

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