A006498 -id:A006498 - cp's OEIS frontend

A157897 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 3, 1, 2, 0, 1, 4, 3, 3, 2, 0, 1, 5, 6, 5, 6, 0, 1, 1, 6, 10, 9, 12, 3, 3, 0, 1, 7, 15, 16, 21, 12, 6, 3, 0, 1, 8, 21, 27, 35, 30, 14, 12, 0, 1, 1, 9, 28, 43, 57, 61, 35, 30, 6, 4, 0, 1, 10, 36, 65, 91, 111, 81, 65, 30, 10, 4, 0, 1, 11, 45, 94, 142, 189, 169, 135, 90, 30, 20, 0, 1

Offset: 0

Gary W. Adamson, Mar 08 2009

T(n, k) is the number of tilings of an n-board that use k (1/2, 1)-fences and n-k squares. A (1/2, 1)-fence is a tile composed of two pieces of width 1/2 separated by a gap of width 1. (Result proved in paper by K. Edwards - see the links section.) - Michael A. Allen, Apr 28 2019

T(n, k) is the (n, n-k)-th entry in the (1/(1-x^3), x*(1+x)/(1-x^3)) Riordan array. - Michael A. Allen, Mar 11 2021

			First few rows of the triangle are:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  0,  1;
  1,  3,  1,  2,  0;
  1,  4,  3,  3,  2,  0;
  1,  5,  6,  5,  6,  0,  1;
  1,  6, 10,  9, 12,  3,  3,  0;
  1,  7, 15, 16, 21, 12,  6,  3,  0;
  1,  8, 21, 27, 35, 30, 14, 12,  0,  1;
  ...
T(9,3) = 27 = T(8,3) + T(7,2) + T(6,0) = 16 + 10 + 1.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
K. Edwards, A Pascal-like triangle related to the tribonacci numbers, Fib. Q., 46/47 (2008/2009), 18-25.
Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.

Cf. A000073 (row sums), A006498, A120415.
Cf. A001477, A079978, A087508, A120415, A161680.
Other triangles related to tiling using fences: A059259, A123521, A335964.

function T(n,k) // T = A157897
  if k lt 0 or k gt n then return 0;
  elif k eq 0 then return 1;
  else return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 01 2022

T[n_,k_]:= If[nMichael A. Allen, Apr 28 2019 *)

def T(n,k): # T = A157897
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    else: return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3)
flatten([[T(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Sep 01 2022

T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

Sum_{k=0..n} T(n, k) = A000073(n+2). - Reinhard Zumkeller, Jun 25 2009

From G. C. Greubel, Sep 01 2022: (Start)

T(n, k) = T(n-1, k) + T(n-2, k-1) + T(n-3, k-3), with T(n, 0) = 1.

T(n, n) = A079978(n).

T(n, n-1) = A087508(n), n >= 1.

T(n, 1) = A001477(n-1).

T(n, 2) = A161680(n-2).

Sum_{k=0..floor(n/2)} T(n-k, k) = A120415(n). (End)

Name clarified by Michael A. Allen, Apr 28 2019

Definition improved by Michael A. Allen, Mar 11 2021

A013979 Expansion of 1/(1 - x^2 - x^3 - x^4) = 1/((1 + x)*(1 - x - x^3)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 17, 24, 36, 52, 77, 112, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 5136, 7528, 11032, 16169, 23696, 34729, 50897, 74594, 109322, 160220, 234813, 344136, 504355, 739169, 1083304, 1587660, 2326828, 3410133, 4997792, 7324621

Offset: 0

N. J. A. Sloane

For n>0, number of compositions (ordered partitions) of n into 2's, 3's and 4's. - Len Smiley, May 08 2001
Diagonal sums of trinomial triangle A071675 (Riordan array (1, x*(1+x+x^2))). - Paul Barry, Feb 15 2005
For n>1, a(n) is number of compositions of n-2 into parts 1 and 2 with no 3 consecutive 1's. For example: a(7) = 5 because we have: 2+2+1, 2+1+2, 1+2+2, 1+2+1+1, 1+1+2+1. - Geoffrey Critzer, Mar 15 2014
In the same way [per 2nd comment for A006498, by Sreyas Srinivasan] that the sum of any two alternating terms (terms separated by one term) of A006498 produces a term from A000045 (the Fibonacci sequence), so it could therefore be thought of as a "metaFibonacci," the sum of any two (nonalternating) terms of this sequence produces a term from A000930 (Narayana’s cows), so this sequence could analogously be called "meta-Narayana’s cows" (e.g. 4+5=9, 5+8=13, 8+11=19, 11+17=28). - Michael Cohen and Yasuyuki Kachi, Jun 13 2024

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 5*x^7 + 8*x^8 + 11*x^9 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael Cohen and Yasuyuki Kachi, Recurrence Relations Rhythm. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol. 14639. Springer, Cham.
C. K. Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), no. 3, 175-189.
C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^0_n.]
R. Mullen, On Determining Paint by Numbers Puzzles with Nonunique Solutions, JIS 12 (2009) 09.6.5.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1).

Cf. A060945 (Ordered partitions into 1's, 2's and 4's).
First differences of A023435.
Cf. A000045, A000930, A001634, A006498, A071675, A077889, A078012, A097333, A107458.

a013979 n = a013979_list !! n
a013979_list = 1 : 0 : 1 : 1 : zipWith (+) a013979_list
   (zipWith (+) (tail a013979_list) (drop 2 a013979_list))
-- Reinhard Zumkeller, Mar 23 2012

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1+x)*(1-x-x^3)) )); // G. C. Greubel, Jul 17 2023

a[n_]:= If[n<0, SeriesCoefficient[x^4/(1 +x +x^2 -x^4), {x, 0, -n}], SeriesCoefficient[1/(1 -x^2 -x^3 -x^4), {x,0,n}]]; (* Michael Somos, Jun 20 2015 *)
LinearRecurrence[{0,1,1,1}, {1,0,1,1}, 50] (* G. C. Greubel, Jul 17 2023 *)

@CachedFunction
def b(n): return 1 if (n<3) else b(n-1) + b(n-3) # b = A000930
def A013979(n): return ((-1)^n +2*b(n) -b(n-1) +b(n-2) -int(n==1))/3
[A013979(n) for n in (0..50)] # G. C. Greubel, Jul 17 2023

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..floor(n/2)} C(k, 2i+3k-n)*C(2i+3k-n, i). - Paul Barry, Feb 15 2005
a(n) = a(n-4) + a(n-3) + a(n-2). - Jon E. Schoenfield, Aug 07 2006
a(n) + a(n+1) = A000930(n+1). - R. J. Mathar, Mar 14 2011
a(n) = (1/3)*(A000930(n) + A097333(n-2) + (-1)^n), n>1. - Ralf Stephan, Aug 15 2013
a(n) = (-1)^n * A077889(-4-n) = A107458(n+4) for all n in Z. - Michael Somos, Jun 20 2015
a(n) = Sum_{i=0..floor(n/2)} A078012(n-2*i). - Paul Curtz, Aug 18 2021
a(n) = (1/3)*((-1)^n + 2*b(n) - b(n-1) + b(n-2) - [n=1]), where b(n) = A000930(n). - G. C. Greubel, Jul 17 2023

A207426 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 114, 81, 14, 25, 225, 361, 351, 196, 21, 40, 625, 1425, 1521, 1162, 441, 31, 64, 1600, 5625, 8463, 6889, 3633, 961, 46, 104, 4096, 20550, 47089, 55361, 29929, 11067, 2116, 68, 169, 10816, 75076, 241087, 444889, 341329

Offset: 1

R. H. Hardin Feb 17 2012

Table starts
..2....4.....6......9.......15........25.........40...........64...........104
..4...16....36.....81......225.......625.......1600.........4096.........10816
..6...36...114....361.....1425......5625......20550........75076........282494
..9...81...351...1521.....8463.....47089.....241087......1234321.......6520459
.14..196..1162...6889....55361....444889....3210938.....23174596.....174570082
.21..441..3633..29929...341329...3892729...39698733....404854641....4315250265
.31..961.11067.127449..2048823..32936121..474552171...6837470721..102807481767
.46.2116.33994.546121.12436631.283215241.5755114104.116947584576.2485504480392

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

R. H. Hardin, Table of n, a(n) for n = 1..365

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)

A207729 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 78, 81, 14, 25, 225, 169, 189, 196, 21, 40, 625, 611, 441, 490, 441, 31, 64, 1600, 2209, 2163, 1225, 1113, 961, 46, 104, 4096, 6016, 10609, 8575, 2809, 2449, 2116, 68, 169, 10816, 16384, 33063, 60025, 27931, 6241, 5474, 4624

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2....4....6.....9.....15......25.......40.......64........104.........169
..4...16...36....81....225.....625.....1600.....4096......10816.......28561
..6...36...78...169....611....2209.....6016....16384......51840......164025
..9...81..189...441...2163...10609....33063...103041.....418263.....1697809
.14..196..490..1225...8575...60025...211680...746496....4078944....22287841
.21..441.1113..2809..27931..277729..1029231..3814209...27996255...205492225
.31..961.2449..6241..88243.1247689..4799749.18464209..184732327..1848226081
.46.2116.5474.14161.288813.5890329.23473944.93547584.1307760792.18282014521

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

R. H. Hardin, Table of n, a(n) for n = 1..649

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)

A207693 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 60, 81, 13, 25, 225, 100, 144, 169, 18, 40, 625, 240, 256, 312, 324, 25, 64, 1600, 576, 768, 576, 612, 625, 34, 104, 4096, 1296, 2304, 1872, 1156, 1250, 1156, 46, 169, 10816, 2916, 5856, 6084, 4216, 2500, 2516, 2116, 62, 273

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2....4....6....9....15.....25.....40......64.....104......169.......273
..4...16...36...81...225....625...1600....4096...10816....28561.....74529
..6...36...60..100...240....576...1296....2916....6804....15876.....36288
..9...81..144..256...768...2304...5856...14884...42700...122500....320600
.13..169..312..576..1872...6084..18564...56644..177548...556516...1724752
.18..324..612.1156..4216..15376..50096..163216..578528..2050624...6804864
.25..625.1250.2500.10000..40000.145200..527076.2056032..8020224..29854944
.34.1156.2516.5476.24420.108900.453420.1887876.8263236.36168196.155101060

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

R. H. Hardin, Table of n, a(n) for n = 1..648

Column 1 is A171861(n+1)
Column 2 is A207025
Column 3 is A207584
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)

A207908 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 102, 81, 13, 25, 225, 289, 261, 169, 18, 40, 625, 1071, 841, 611, 324, 25, 64, 1600, 3969, 4089, 2209, 1278, 625, 34, 104, 4096, 13230, 19881, 13865, 5041, 2625, 1156, 46, 169, 10816, 44100, 80511, 87025, 39831, 11025

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2....4....6.....9.....15......25.......40........64........104.........169
..4...16...36....81....225.....625.....1600......4096......10816.......28561
..6...36..102...289...1071....3969....13230.....44100.....153090......531441
..9...81..261...841...4089...19881....80511....326041....1428071.....6255001
.13..169..611..2209..13865...87025...417425...2002225...10896915....59305401
.18..324.1278..5041..39831..314721..1726758...9474084...62431074...411400089
.25..625.2625.11025.110775.1113025..6835345..41977441..336253621..2693506201
.34.1156.5134.22801.289467.3674889.24832818.167806116.1627138986.15777620881

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

R. H. Hardin, Table of n, a(n) for n = 1..513

Column 1 is A171861(n+1)
Column 2 is A207025
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207704

A207928 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 15, 81, 72, 100, 16, 25, 225, 144, 240, 256, 26, 40, 625, 360, 576, 704, 676, 42, 64, 1600, 900, 1872, 1936, 2080, 1764, 68, 104, 4096, 2160, 6084, 7744, 6400, 6216, 4624, 110, 169, 10816, 5184, 18096, 30976, 29760, 21904

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2....4.....6.....9.....15......25.......40........64.......104........169
..4...16....36....81....225.....625.....1600......4096.....10816......28561
..6...36....72...144....360.....900.....2160......5184.....12528......30276
.10..100...240...576...1872....6084....18096.....53824....165648.....509796
.16..256...704..1936...7744...30976...111584....401956...1513992....5702544
.26..676..2080..6400..29760..138384...592968...2540836..11151624...48944016
.42.1764..6216.21904.126688..732736..3773248..19430464.105642128..574369156
.68.4624.18496.73984.520608.3663396.22906752.143233024.955190016.6369955344

			Some solutions for n=9 k=3
..0..0..1....0..0..1....0..1..1....1..0..0....1..0..0....0..0..1....0..0..1
..0..1..1....1..1..1....0..0..1....1..0..0....0..0..1....1..1..1....0..0..1
..1..1..0....1..0..0....1..0..0....0..1..1....1..1..0....1..1..0....1..1..0
..1..0..0....0..1..1....0..1..1....0..0..1....0..1..1....0..0..1....0..1..1
..0..1..1....1..1..0....1..1..1....1..1..0....0..0..1....1..0..1....1..0..0
..0..1..1....0..0..1....1..0..0....1..0..0....1..1..0....1..1..0....0..1..1
..1..0..0....1..0..0....0..1..1....0..1..1....0..1..1....0..1..1....1..0..1
..1..1..0....1..1..1....1..1..1....1..0..0....0..0..1....0..0..1....1..0..0
..0..1..1....0..1..1....1..0..0....1..1..1....1..0..0....1..1..0....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..391

Column 1 is A006355(n+2)
Column 2 is A206981
Column 3 is A207840
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)

A208108 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 78, 81, 14, 25, 225, 169, 171, 196, 22, 40, 625, 611, 361, 406, 484, 35, 64, 1600, 2209, 1805, 841, 990, 1225, 56, 104, 4096, 6016, 9025, 6235, 2025, 2485, 3136, 90, 169, 10816, 16384, 25555, 46225, 22995, 5041, 6328, 8100

Offset: 1

R. H. Hardin Feb 23 2012

Table starts
..2....4....6.....9.....15.......25.......40.......64........104.........169
..4...16...36....81....225......625.....1600.....4096......10816.......28561
..6...36...78...169....611.....2209.....6016....16384......51840......164025
..9...81..171...361...1805.....9025....25555....72361.....292941.....1185921
.14..196..406...841...6235....46225...134160...389376....2189616....12313081
.22..484..990..2025..22995...261121...768544..2262016...18608992...153091129
.35.1225.2485..5041..89815..1600225..4747545.14085009..176312187..2207026441
.56.3136.6328.12769.361261.10220809.30461016.90782784.1769397240.34486347025

			Some solutions for n=10 k=3
..1..0..0....0..0..1....1..0..1....1..1..0....1..0..0....0..1..1....0..0..1
..0..0..1....1..0..0....0..1..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..1..1....0..0..1....1..1..0....1..1..0....0..1..1....1..0..1
..0..1..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..1..1....0..1..1....1..1..0....1..0..0....0..1..1....1..0..1
..0..1..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..0..1....1..1..1....1..1..0....1..0..0....0..1..1....1..1..1
..0..0..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..0..1
..1..1..0....0..1..1....1..1..1....1..1..0....1..0..0....0..1..1....0..0..1
..0..0..1....1..0..0....0..0..1....0..1..1....0..0..1....1..1..0....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..845

Column 1 is A001611(n+2)
Column 2 is A207436
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207730

A335964 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-3,k-1) + T(n-4,k-2) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 0, 1, 4, 4, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 1, 6, 11, 6, 1, 0, 0, 0, 0, 1, 7, 16, 13, 3, 0, 0, 0, 0, 0, 1, 8, 22, 24, 9, 0, 0, 0, 0, 0, 0, 1, 9, 29, 40, 22, 3, 0, 0, 0, 0, 0, 0

Offset: 0

Michael A. Allen, Jul 01 2020

T(n,k) is the number of tilings of an n-board (a board with dimensions n X 1) using k (1,1)-fence tiles and n-2k square tiles. A (w,g)-fence tile is composed of two tiles of width w separated by a gap of width g.
Sum of n-th row = A006498(n).
T(2*j+r,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x) = f(n,x) + x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0. - Michael A. Allen, Oct 02 2021

			Triangle begins:
  1;
  1,  0;
  1,  0,  0;
  1,  1,  0,  0;
  1,  2,  1,  0,  0;
  1,  3,  2,  0,  0,  0;
  1,  4,  4,  0,  0,  0,  0;
  1,  5,  7,  2,  0,  0,  0,  0;
  1,  6, 11,  6,  1,  0,  0,  0,  0;
  1,  7, 16, 13,  3,  0,  0,  0,  0,  0;
  1,  8, 22, 24,  9,  0,  0,  0,  0,  0,  0;
  1,  9, 29, 40, 22,  3,  0,  0,  0,  0,  0,  0;
  ...

Kenneth Edwards and Michael A. Allen, New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile, arXiv:2009.04649 [math.CO], 2020.
Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.
John Konvalina, On the number of combinations without unit separation., Journal of Combinatorial Theory, Series A 31.2 (1981): 101-107. See Table I.

Other triangles related to tiling using fences: A059259, A123521, A157897, A158909.

Cf. A006498 (row sums), A011973, A348445.

Mathematica
```
T[n_,k_]:=If[n
				
```

TT(n,k) = if (nA059259
T(n,k) = TT(n-k,k);
\\ matrix(7,7,n,k, T(n-1,k-1)) \\ Michel Marcus, Jul 18 2020

T(n,k) = A059259(n-k,k).
From Michael A. Allen, Oct 02 2021: (Start)
G.f.: 1/((1 + x^2*y)(1 - x - x^2*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the expansion of the g.f.
T(n,0) = 1.
T(n,1) = n-2 for n>1.
T(n,2) = binomial(n-4,2) + n - 3 for n>3.
T(n,3) = binomial(n-6,3) + 2*binomial(n-5,2) for n>5.
T(4*m-3,2*m-2) = T(4*m-1,2*m-1) = m for m>0.
T(2*n+1,n-k) = A158909(n,k). (End)
T(n,k) = A348445(n-2,k) for n>1.

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A264379 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 2,-2 -1,0 -1,2 1,0 or 0,-1.

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A157897 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

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A013979 Expansion of 1/(1 - x^2 - x^3 - x^4) = 1/((1 + x)*(1 - x - x^3)).

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A207426 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 0 vertically.

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A207729 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207693 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 1 1 vertically.

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A207908 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 0 1 1 vertically.

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A207928 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 1 1 1 vertically.

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A208108 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 1 0 vertically.

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