A052912 -id:A052912 - cp's OEIS frontend

A239819 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it, modulo 4.

2, 4, 5, 10, 23, 11, 24, 132, 113, 25, 56, 729, 1480, 582, 57, 132, 3951, 18728, 17552, 2981, 129, 312, 21602, 232272, 510748, 204779, 15266, 293, 736, 118253, 2912793, 14544801, 13597573, 2405330, 78188, 665, 1736, 646306, 36627126, 418324402

Offset: 1

R. H. Hardin, Mar 27 2014

Table starts
....2.......4.........10............24................56..................132
....5......23........132...........729..............3951................21602
...11.....113.......1480.........18728............232272..............2912793
...25.....582......17552........510748..........14544801............418324402
...57....2981.....204779......13597573.........884977259..........58232200212
..129...15266....2405330.....366379173.......54668820459........8243207656791
..293...78188...28156167....9807771898.....3347474694032.....1154988223050638
..665..400542..330152684..263419973152...205970817822022...162794110794893005
.1509.2051667.3868656623.7064275271994.12641836066488239.22871029907841066549

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

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

Row 1 is A052912

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) +2*a(n-3)
k=2: [order 10]
k=3: [order 35]
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-3)
n=2: [order 16]
n=3: [order 64]

A241283 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or zero plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

Original entry on oeis.org

2, 3, 4, 4, 3, 10, 7, 5, 9, 24, 10, 13, 41, 36, 56, 15, 17, 126, 236, 139, 132, 24, 35, 224, 773, 1615, 532, 312, 35, 90, 934, 1800, 6783, 12356, 2111, 736, 54, 141, 2741, 16843, 20717, 77955, 96171, 8473, 1736, 83, 288, 5225, 54167, 451318, 309657, 1009773

Offset: 1

R. H. Hardin, Apr 18 2014

Table starts
....2......3........4..........7..........10............15...........24
....4......3........5.........13..........17............35...........90
...10......9.......41........126.........224...........934.........2741
...24.....36......236........773........1800.........16843........54167
...56....139.....1615.......6783.......20717........451318......1543713
..132....532....12356......77955......309657......15616828.....60486552
..312...2111....96171....1009773.....5423164.....730435588...2949336562
..736...8473...761754...14440961...113305157...40083129145.193899487190
.1736..34053..6079503..217830879..2759021846.2390612565177
.4096.136880.48655224.3381893022.75062814060

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A052912

Row 1 is A159288(n+1)

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-3)
k=2: [order 31]
Empirical for row n:
n=1: a(n)=a(n-2)+2*a(n-3)
n=2: [order 17] for n>18

A239599 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

Original entry on oeis.org

2, 4, 3, 10, 3, 4, 24, 15, 4, 7, 56, 64, 31, 4, 10, 132, 244, 187, 82, 5, 15, 312, 1030, 1310, 643, 177, 7, 24, 736, 4303, 9806, 8773, 1737, 458, 8, 35, 1736, 17923, 76769, 128347, 38824, 7461, 1071, 11, 54, 4096, 75264, 611126, 1991329, 1031560, 282333, 24946

Offset: 1

R. H. Hardin, Mar 22 2014

Table starts
..2..4....10.....24.......56........132.........312.........736........1736
..3..3....15.....64......244.......1030........4303.......17923.......75264
..4..4....31....187.....1310.......9806.......76769......611126.....4929897
..7..4....82....643.....8773.....128347.....1991329....31686730...510551001
.10..5...177...1737....38824....1031560....30460289...944810972.30050150163
.15..7...458...7461...282333...12509870...681399736.40483561185
.24..8..1071..24946..1583770..120511910.11135057785
.35.11..2150..78667..8002162.1104844380
.54.12..5209.313003.56967196
.83.16.11204.946740

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

R. H. Hardin, Table of n, a(n) for n = 1..95

Column 1 is A159288(n+1)

Row 1 is A052912

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: a(n) = a(n-2) +a(n-5) for n>6
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-3)
n=2: [order 44] for n>47

A240271 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one 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, 3, 10, 7, 4, 24, 35, 14, 7, 56, 157, 118, 36, 10, 132, 713, 919, 582, 72, 15, 312, 3263, 7562, 8265, 2000, 170, 24, 736, 14895, 64721, 126286, 49921, 8353, 411, 35, 1736, 68101, 563496, 2059061, 1363144, 382690, 37422, 879, 54, 4096, 311509, 4956889

Offset: 1

R. H. Hardin, Apr 03 2014

Table starts
..2....4......10.........24..........56..........132...........312
..3....7......35........157.........713.........3263.........14895
..4...14.....118........919........7562........64721........563496
..7...36.....582.......8265......126286......2059061......34514871
.10...72....2000......49921.....1363144.....40760821....1277623744
.15..170....8353.....382690....19210586...1063706501...63085436203
.24..411...37422....3076452...278945445..27923918285.3004792552569
.35..879..135463...19781372..3200032085.576407548906
.54.2106..580528..154994425.46095401280
.83.4874.2403439.1144262410

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

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A159288(n+1)

Row 1 is A052912

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 13]
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-3)
n=2: [order 26] for n>28

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

A089978 Expansion of 1/(1-3x-3x^3).

Original entry on oeis.org

1, 3, 9, 30, 99, 324, 1062, 3483, 11421, 37449, 122796, 402651, 1320300, 4329288, 14195817, 46548351, 152632917, 500486202, 1641103659, 5381209728, 17645087790, 57858574347, 189719352225, 622093320045, 2039855683176

Offset: 0

Paul Barry, Nov 18 2003

			G.f. = 1 + 3*x + 9*x^2 + 30*x^3 + 99*x^4 + 324*x^5 + 1062*x^6 + 3483*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,3).

Cf. A052912, A000930.

CoefficientList[Series[1/(1-3x-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {3,0,3},{1,3,9},30] (* Harvey P. Dale, Jul 22 2015 *)

{a(n) = sum(k=0, n\3, binomial(n - 2*k, k) * 3^(n - 2*k))}; /* Michael Somos, Jan 30 2015 */

{a(n) = if( n<0, 0, polcoeff( 1 / (1 - 3*x - 3*x^3) + x * O(x^n), n))}; /* Michael Somos, Jan 30 2015 */

a(n) = Sum_{k=0..floor(n/2)} C(n-2*k, k)3^(n-2*k).

a(0)=1, a(1)=3, a(2)=9, a(n)=3*a(n-1)+3*a(n-3). - Harvey P. Dale, Jul 22 2015

A089979 Expansion of 1/(1-4x-4x^3).

Original entry on oeis.org

1, 4, 16, 68, 288, 1216, 5136, 21696, 91648, 387136, 1635328, 6907904, 29180160, 123261952, 520679424, 2199438336, 9290801152, 39245922304, 165781442560, 700288974848, 2958139588608, 12495684124672, 52783892398080

Offset: 0

Paul Barry, Nov 18 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,4).

Cf. A052912, A000930, A089978.

CoefficientList[Series[1/(1 - 4*x - 4*x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{4,0,4}, {1,4,16}, 50] (* G. C. Greubel, Apr 29 2017 *)

x='x+O('x^50); Vec(1/(1 - 4*x - 4*x^3)) \\ G. C. Greubel, Apr 29 2017

a(n) = Sum_{k=0..floor(n/2)} C(n-2*k, k)*4^(n-2*k).

a(n) = 4*a(n-1) + 4*a(n-3). - G. C. Greubel, Apr 29 2017

A077999 Expansion of (1-x)/(1-2x-2x^3).

Original entry on oeis.org

1, 1, 2, 6, 14, 32, 76, 180, 424, 1000, 2360, 5568, 13136, 30992, 73120, 172512, 407008, 960256, 2265536, 5345088, 12610688, 29752448, 70195072, 165611520, 390727936, 921846016, 2174915072, 5131286016, 12106264064, 28562358272, 67387288576, 158987105280

Offset: 0

N. J. A. Sloane, Nov 17 2002

a(n) = number of permutations on [n] that avoid nonconsecutive instances of the patterns 321 and 312. For example, a(4) does not count pi=4231 because 431 forms a 321 pattern in pi but 431 is not a consecutive (that is, contiguous) string in pi; also, the first 3 letters form a 312 pattern but that's not disqualifying because they do occur consecutively. Counting these permutations by various statistics yields the listed formulas/recurrences. - David Callan, Oct 26 2006

a(n) = term (1,1) of M^n, M = the 4 X 4 matrix [1,0,1,1; 1,1,0,0; 0,1,0,1; 1,0,0,1]. a(n)/a(n-1) tends to 2.3593040859..., an eigenvalue of the matrix and a root to the characteristic polynomial x^4 - 3x^3 + 2x^2 - 2x + 2. - Gary W. Adamson, Oct 01 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. R. Finch, Several Constants Arising in Statistical Mechanics, arXiv:math/9810155 [math.CO], 1998-1999. see p. 8.
Index entries for linear recurrences with constant coefficients, signature (2,0,2)

Cf. A052912, A110524.

a:=[1,1,2];; for n in [4..40] do a[n]:=2*(a[n-1]+a[n-3]); od; a; # G. C. Greubel, Jun 27 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/( 1-2*x-2*x^3) )); // G. C. Greubel, Jun 27 2019

CoefficientList[Series[(1-x)/(1-2x-2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,2},{1,1,2},40] (* Harvey P. Dale, Sep 10 2016 *)

my(x='x+O('x^40)); Vec((1-x)/(1-2*x-2*x^3)) \\ G. C. Greubel, Jun 27 2019

((1-x)/(1-2*x-2*x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

a(n) = 2*(a(n-1) + a(n-3)) counts the above permutations by first entry. a(n) = a(n-1) + a(n-2) + 3*Sum_{k=0..n-3} a(k) counts by last entry. a(n) = 2^(n-1) + Sum_{k=0..n-3} 2^(n-2-k)*a(k) counts by location of first 3xx pattern. a(n) = Sum_{k=0..floor(n/3)} ((n-k)/(n-2k))* binomial(n-2*k,k) * 2^(n-2*k-1) counts by number of 3xx patterns. - David Callan, Oct 26 2006
a(n) = A052912(n) - A052912(n-1). - R. J. Mathar, May 30 2014
a(n) = (-1)^n * A110524(n). - G. C. Greubel, Jun 27 2019

A077851 Expansion of (1-x)^(-1)/(1 - 2x - 2x^3).

Original entry on oeis.org

1, 3, 7, 17, 41, 97, 229, 541, 1277, 3013, 7109, 16773, 39573, 93365, 220277, 519701, 1226133, 2892821, 6825045, 16102357, 37990357, 89630805, 211466325, 498913365, 1177088341, 2777109333, 6552045397, 15458267477, 36470753621, 86045598037, 203007731029

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,-2,2,-2).

A052912 := proc(n) if n < 0 then 0; else coeftayl( 1/(1-2*x^3-2*x),x=0,n) ; end if; end proc:
A077851 := proc(n) 2*(2*A052912(n)+A052912(n-1)+A052912(n-2))-1 ;  %/3 ; end proc:
seq(A077851(n),n=0..15) ; # R. J. Mathar, Mar 28 2011

Vec((1-x)^(-1)/(1-2*x-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A077927 Expansion of (1-x)^(-1)/(1+2x+2x^3).

Original entry on oeis.org

1, -1, 3, -7, 17, -39, 93, -219, 517, -1219, 2877, -6787, 16013, -37779, 89133, -210291, 496141, -1170547, 2761677, -6515635, 15372365, -36268083, 85567437, -201879603, 476295373, -1123725619, 2651210445, -6255011635, 14757474509, -34817369907, 82144763085, -193804475187

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (-1,2,-2,2).

a(n) = -a(n-1) + 2 a(n-2) - 2 a(n-3) + 2 a(n-4) - N-E. Fahssi, Mar 29 2008

a(n) -a(n-1) = (-1)^(n+1)*A052912(n). - R. J. Mathar, Jul 08 2022

cp's OEIS Frontend

A239819 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it, modulo 4.

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A241283 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or zero plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

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A239599 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4.

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A240271 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one 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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Mathematica

PARI

Formula

A089978 Expansion of 1/(1-3x-3x^3).

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Mathematica

PARI

PARI

Formula

A089979 Expansion of 1/(1-4x-4x^3).

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Programs

Mathematica

PARI

Formula

A077999 Expansion of (1-x)/(1-2*x-2*x^3).

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GAP

Magma

A077999 Expansion of (1-x)/(1-2x-2x^3).

A077851 Expansion of (1-x)^(-1)/(1 - 2x - 2x^3).

A077927 Expansion of (1-x)^(-1)/(1+2x+2x^3).