A239340 -id:A239340 - cp's OEIS frontend

A239333 Number of n X 1 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

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

Offset: 1

R. H. Hardin, Mar 16 2014

nonn

Column 1 of A239340.

			Some solutions for n=5:
..0....2....2....2....0....0....0....0....2....2....2....2....2....0....2....2
..2....0....2....0....2....0....2....0....0....0....0....2....2....2....0....3
..2....2....2....3....2....2....3....0....2....2....3....0....0....0....3....2
..0....0....0....2....2....2....2....0....2....2....3....0....2....0....2....3
..2....2....2....0....2....0....0....0....3....0....2....2....3....2....3....2

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 2*a(n-1) +2*a(n-3).

Empirical g.f.: (2 + x + 2*x^2) / (1 - 2*x - 2*x^3). - Colin Barker, Feb 18 2018

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

A239334 Number of nX2 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

Original entry on oeis.org

5, 31, 172, 926, 5078, 27861, 152260, 832207, 4550810, 24881731, 136029644, 743697760, 4065948624, 22229220072, 121530709245, 664428372582, 3632538661095, 19859679158791, 108576093277449, 593603153333233

Offset: 1

R. H. Hardin, Mar 16 2014

nonn

Column 2 of A239340

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 9*a(n-1) -28*a(n-2) +74*a(n-3) -181*a(n-4) +236*a(n-5) -261*a(n-6) +234*a(n-7) +200*a(n-8) -279*a(n-9) +104*a(n-10) -268*a(n-11) -36*a(n-12) +156*a(n-13) +104*a(n-14) -32*a(n-15) -16*a(n-16)

A239335 Number of nX3 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

Original entry on oeis.org

12, 172, 2187, 27341, 346028, 4360887, 54774178, 688266175, 8649330696, 108663198449, 1365050486095, 17148274201252, 215420739233431, 2706135761424448, 33994663679676011, 427043158643798758

Offset: 1

R. H. Hardin, Mar 16 2014

nonn

Column 3 of A239340

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

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 64

Empirical recurrence of order 64 (see link above)

A239336 Number of nX4 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

Original entry on oeis.org

28, 926, 27341, 790681, 22952323, 663249556, 19117551262, 550960518608, 15873533676775, 457181997376959, 13166310184133045, 379162593435951456, 10918763117793096685, 314422821700641922733, 9054241291311457562833

Offset: 1

R. H. Hardin, Mar 16 2014

nonn

Column 4 of A239340

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

R. H. Hardin, Table of n, a(n) for n = 1..210

A239337 Number of nX5 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

Original entry on oeis.org

66, 5078, 346028, 22952323, 1527488498, 101276030555, 6693244281896, 442014330398515, 29178369838713161, 1925440562309496640, 127035121924736497404, 8380770923086769177166, 552870412078787577689020

Offset: 1

R. H. Hardin, Mar 16 2014

nonn

Column 5 of A239340

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

R. H. Hardin, Table of n, a(n) for n = 1..210

A239338 Number of nX6 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

Original entry on oeis.org

156, 27861, 4360887, 663249556, 101276030555, 15397080750829, 2331550702010844, 352742924096563362, 53342694175522093768, 8062792674253914737279, 1218391421349788433734529

Offset: 1

R. H. Hardin, Mar 16 2014

nonn

Column 6 of A239340

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

R. H. Hardin, Table of n, a(n) for n = 1..210

cp's OEIS Frontend

A239333 Number of n X 1 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, 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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A239334 Number of nX2 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

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A239335 Number of nX3 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

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A239336 Number of nX4 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

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A239337 Number of nX5 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

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