A078039 -id:A078039 - cp's OEIS frontend

A002478 Bisection of A000930.

1, 1, 3, 6, 13, 28, 60, 129, 277, 595, 1278, 2745, 5896, 12664, 27201, 58425, 125491, 269542, 578949, 1243524, 2670964, 5736961, 12322413, 26467299, 56849086, 122106097, 262271568, 563332848, 1209982081, 2598919345, 5582216355, 11990037126, 25753389181

Offset: 0

N. J. A. Sloane

Number of ways to tile a 3 X n region with 1 X 1, 2 X 2 and 3 X 3 tiles.
Number of ternary words with subwords (0,0), (0,1) and (1,1) not allowed. - Olivier Gérard, Aug 28 2012
Diagonal sums of A063967. - Paul Barry, Nov 09 2005
Row sums of number triangle A116088. - Paul Barry, Feb 04 2006
Sequence is identical to its second differences negated, minus the first 3 terms. - Paul Curtz, Feb 10 2008
a(n) = term (3,3) in the 3 X 3 matrix [0,1,0; 0,0,1; 1,2,1]^n. - Gary W. Adamson, May 30 2008
a(n)/a(n-1) tends to 2.147899035..., an eigenvalue of the matrix and a root to x^3 - x^2 - 2x - 1 = 0. - Gary W. Adamson, May 30 2008
INVERT transform of (1, 2, 1, 0, 0, 0, ...) = (1, 3, 6, 13, 28, ...); such that (1, 2, 1, 0, 0, 0, ...) convolved with (1, 1, 3, 6, 13, 28, 0, 0, 0, ...) shifts to the left. - Gary W. Adamson, Apr 18 2010
a(n) is the top left entry in the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 0, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014

			a(3)=6 as there is one tiling of a 3 X 3 region with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 1 tiling consisting of the 3 X 3 tile.

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 322.
S. Heubach, Tiling an m X n Area with Squares of Size up to k X k (m<=5), Congressus Numerantium 140 (1999), pp. 43-64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..300
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Emeric Deutsch, Counting tilings with L-tiles and squares, Problem 10877, Amer. Math. Monthly, 110 (March 2003), 245-246.
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, arXiv:1907.06517 [math.CO], 2019.
Leonhard Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 412
Milan Janjić, Pascal Triangle and Restricted Words, arXiv:1705.02497 [math.CO], 2017.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Richard J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 19 (halved...).
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 5.
Sam Northshield, Some generalizations of a formula of Reznick, SUNY Plattsburgh (2022).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Cf. A000930, A054856, A054857, A025234, A078007, A078039, A226546, A077936 (INVERT transform), A008346 (inverse INVERT transform).

I:=[1,1,3]; [n le 3 select I[n] else Self(n-1) +2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 14 2023

f[A_]:= Module[{til = A}, AppendTo[til, A[[-1]] + 2A[[-2]] + A[[-3]]]]; NumOfTilings[ n_ ]:= Nest[ f, {1,1,3}, n-2]; NumOfTilings[30]
LinearRecurrence[{1,2,1},{1,1,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
CoefficientList[Series[1/(1-x-2x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Oct 17 2024 *)

a(n)=([0,1,0; 0,0,1; 1,2,1]^n*[1;1;3])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

@CachedFunction
def a(n): # A002478
    if (n<3): return (1,1,3)[n]
    else: return sum(binomial(2,j)*a(n-j) for j in range(1,4))
[a(n) for n in (0..40)] # G. C. Greubel, Apr 14 2023

G.f.: 1 / (1-x-2*x^2-x^3). [Simon Plouffe in his 1992 dissertation.]
a(n) = a(n-1) + 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} binomial(2*n-2*k, k). - Paul Barry, Nov 13 2004
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j, n-k-j)*C(j, k). - Paul Barry, Nov 09 2005
a(n) = Sum_{k=0..n} C(2*k,n-k) = Sum_{k=0..n} C(n,k)*C(3*k,n)/C(3*k,k). - Paul Barry, Feb 04 2006
a(n) = A000930(n) + 2*Sum_{i=0..n-2} a(i)*A000930(n-2-i). - Michael Tulskikh, Jun 07 2020

Additional comments from Silvia Heubach (silvi(AT)cine.net), Apr 21 2000

A297224 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 1 neighboring 1s.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 8, 9, 1, 6, 16, 24, 19, 1, 9, 33, 57, 68, 41, 1, 13, 69, 182, 207, 196, 88, 1, 19, 145, 535, 997, 751, 564, 189, 1, 28, 300, 1513, 4210, 5570, 2720, 1620, 406, 1, 41, 624, 4415, 16658, 33158, 30946, 9861, 4660, 872, 1, 60, 1300, 12832, 68769, 178469

Offset: 1

R. H. Hardin, Dec 27 2017

Table starts
.1...2.....3......4.......6.........9.........13..........19............28
.1...4.....8.....16......33........69........145.........300...........624
.1...9....24.....57.....182.......535.......1513........4415.........12832
.1..19....68....207.....997......4210......16658.......68769........284867
.1..41...196....751....5570.....33158.....178469.....1051514.......6152761
.1..88...564...2720...30946....261939....1918732....16176806.....134671502
.1.189..1620...9861..171851...2063378...20599895...248421807....2936448567
.1.406..4660..35741..955316..16277793..221333623..3819208252...64142817874
.1.872.13396.129540.5308160.128351805.2377449633.58680928294.1400212345305

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

R. H. Hardin, Table of n, a(n) for n = 1..611

Column 2 is A078039.
Row 1 is A000930(n+1).
Row 2 is A264166.

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: a(n) = a(n-1) +4*a(n-2) +4*a(n-3)
k=4: a(n) = a(n-1) +6*a(n-2) +11*a(n-3) +6*a(n-4) +a(n-5)
k=5: [order 9]
k=6: [order 11] for n>13
k=7: [order 16] for n>21
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-3)
n=2: a(n) = a(n-1) +a(n-2) +2*a(n-3) +a(n-4) +a(n-5) -a(n-6)
n=3: [order 13]
n=4: [order 27]
n=5: [order 60]

A141015 a(0) = 0, a(1) = 1, a(2) = 2; for n > 2, a(n) = a(n-1) + 2*a(n-2) + a(n-3).

Original entry on oeis.org

0, 1, 2, 4, 9, 19, 41, 88, 189, 406, 872, 1873, 4023, 8641, 18560, 39865, 85626, 183916, 395033, 848491, 1822473, 3914488, 8407925, 18059374, 38789712, 83316385, 178955183, 384377665, 825604416, 1773314929, 3808901426

Offset: 0

Juri-Stepan Gerasimov, Jul 11 2008

Central axis of triangle G(n, k): G(n,0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, G(n+4, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) for k = 1..(n+1). (This is triangular array A140997.)
Central axis of triangle G(n, k): G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, 2) = 4, G(n+4, k) = G(n+1, k-2) + G(n+1, k-3) + G(n+2, k-2) + G(n+3, k-1) for k = 3..(n+3). (This is triangular array A140994, which is a mirror image of A140997.)
a(n-1) is the top left entry of the n-th power of any of the 3X3 matrices [0, 1, 1; 1, 1, 1; 0, 1, 0], [0, 1, 0; 1, 1, 1; 1, 1, 0], [0, 1, 1; 0, 0, 1; 1, 1, 1] or [0, 0, 1; 1, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Cf. A000129, A140994, A140997.

CoefficientList[Series[x (1 + x)/(1 - x - 2 x^2 - x^3), {x,0,50}], x] (* G. C. Greubel, Jun 09 2017 *)

x='x+O('x^50); concat([0], Vec(x*(1+x)/(1-x-2*x^2-x^3))) \\ G. C. Greubel, Jun 09 2017

from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(0,1,2,1,2,1); [next(it) for i in range(31)] # Zerinvary Lajos, May 17 2009

From R. J. Mathar, Aug 22 2008: (Start)
O.g.f.: x*(1 + x)/(1 - x - 2*x^2 - x^3).
a(n) = (-1)^(n+1)*A078039(n-1). (End)

Typo in definition corrected by Paolo P. Lava, Jul 31 2008
Dysfunctional Maple program removed by R. J. Mathar, Oct 28 2009
Comments clarified by Petros Hadjicostas, Jun 12 2019

A228482 T(n,k)=Number of nXk binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 9, 14, 9, 5, 8, 19, 41, 41, 19, 8, 13, 41, 127, 172, 127, 41, 13, 21, 88, 386, 728, 728, 386, 88, 21, 34, 189, 1181, 3084, 4354, 3084, 1181, 189, 34, 55, 406, 3605, 13050, 25699, 25699, 13050, 3605, 406, 55, 89, 872, 11013, 55252, 152373

Offset: 1

R. H. Hardin Aug 22 2013

Table starts
..1...1.....2......3.......5.........8.........13..........21............34
..1...2.....4......9......19........41.........88.........189...........406
..2...4....14.....41.....127.......386.......1181........3605.........11013
..3...9....41....172.....728......3084......13050.......55252........233875
..5..19...127....728....4354.....25699.....152373......902042.......5342712
..8..41...386...3084...25699....211588....1748684....14433982.....119188751
.13..88..1181..13050..152373...1748684...20185842...232542935....2680777055
.21.189..3605..55252..902042..14433982..232542935..3737615288...60122232373
.34.406.11013.233875.5342712.119188751.2680777055.60122232373.1349721589622
Same recurrences as A228285 except in addition this smaller one for k=5

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

R. H. Hardin, Table of n, a(n) for n = 1..1333

Column 1 is A000045
Column 2 is A078039(n-1).
Cf. A228476-A228481, A228277-A228285, A226444.

Recurrences for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: a(n) = a(n-1) +5*a(n-2) +4*a(n-3) -a(n-5)
k=4: a(n) = a(n-1) +10*a(n-2) +15*a(n-3) +4*a(n-4) -6*a(n-5) -a(n-6) +3*a(n-7) -a(n-8)
k=5: a(n) = 3*a(n-1) +15*a(n-2) +16*a(n-3) -11*a(n-4) -20*a(n-5) +19*a(n-6) -8*a(n-7) +a(n-9)
k=6: a(n) = a(n-1) +42*a(n-2) +147*a(n-3) +70*a(n-4) -478*a(n-5) -449*a(n-6) +1199*a(n-7) +732*a(n-8) -2727*a(n-9) +659*a(n-10) +3827*a(n-11) -5776*a(n-12) +3926*a(n-13) -1152*a(n-14) -148*a(n-15) +154*a(n-16) +32*a(n-17) -29*a(n-18) -6*a(n-19) +3*a(n-20) +a(n-21)
k=7: a(n) = a(n-1) +85*a(n-2) +432*a(n-3) +192*a(n-4) -3711*a(n-5) -5096*a(n-6) +21164*a(n-7) +27340*a(n-8) -112654*a(n-9) -37244*a(n-10) +477721*a(n-11) -464722*a(n-12) -897815*a(n-13) +3102284*a(n-14) -4149918*a(n-15) +2761082*a(n-16) -138325*a(n-17) -1353257*a(n-18) +942033*a(n-19) +64683*a(n-20) -365483*a(n-21) +80904*a(n-22) +92350*a(n-23) -27097*a(n-24) -23292*a(n-25) +2585*a(n-26) +5635*a(n-27) +1405*a(n-28) -561*a(n-29) -545*a(n-30) -173*a(n-31) -14*a(n-32) +5*a(n-33) +a(n-34)

A280069 T(n,k)=Number of nXk 0..1 arrays with no element unequal to a strict majority of its horizontal, vertical and antidiagonal neighbors and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, 4, 3, 5, 9, 15, 15, 9, 5, 8, 19, 39, 52, 39, 19, 8, 13, 41, 104, 170, 170, 104, 41, 13, 21, 88, 281, 603, 790, 603, 281, 88, 21, 34, 189, 771, 2157, 3729, 3729, 2157, 771, 189, 34, 55, 406, 2122, 7777, 17468, 23564, 17468, 7777, 2122

Offset: 1

R. H. Hardin, Dec 25 2016

Table starts
..1...1....1......2.......3........5.........8..........13...........21
..1...1....2......4.......9.......19........41..........88..........189
..1...2....5.....15......39......104.......281.........771.........2122
..2...4...15.....52.....170......603......2157........7777........28195
..3...9...39....170.....790.....3729.....17468.......82769.......394904
..5..19..104....603....3729....23564....145485......915505......5786757
..8..41..281...2157...17468...145485...1188839.....9934415.....83159859
.13..88..771...7777...82769...915505...9934415...110266512...1225662273
.21.189.2122..28195..394904..5786757..83159859..1225662273..18113056960
.34.406.5858.102429.1890877.36671797.698377561.13669391105.268403179093

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

R. H. Hardin, Table of n, a(n) for n = 1..263

Column 1 is A000045(n-1).

Column 2 is A078039(n-2).

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2) for n>3
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3) for n>4
k=3: a(n) = 3*a(n-1) +a(n-2) -6*a(n-3) +5*a(n-4) -2*a(n-5) -2*a(n-6) -a(n-7) for n>9
k=4: [order 15] for n>18
k=5: [order 35] for n>40
k=6: [order 87] for n>91

A300465 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3 or 4 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 4, 2, 4, 0, 0, 9, 8, 8, 9, 0, 0, 19, 22, 28, 22, 19, 0, 0, 41, 68, 142, 142, 68, 41, 0, 0, 88, 212, 540, 1146, 540, 212, 88, 0, 0, 189, 652, 2585, 7456, 7456, 2585, 652, 189, 0, 0, 406, 2017, 11343, 55663, 78195, 55663, 11343, 2017, 406, 0, 0

Offset: 1

R. H. Hardin, Mar 06 2018

Table starts
.0...0....0.....0.......0.........0..........0............0..............0
.0...1....2.....4.......9........19.........41...........88............189
.0...2....2.....8......22........68........212..........652...........2017
.0...4....8....28.....142.......540.......2585........11343..........51501
.0...9...22...142....1146......7456......55663.......400028........2906875
.0..19...68...540....7456.....78195.....935121.....10913850......127028268
.0..41..212..2585...55663....935121...17675111....330639367.....6123205255
.0..88..652.11343..400028..10913850..330639367...9829599051...293961266876
.0.189.2017.51501.2906875.127028268.6123205255.293961266876.14061129641181

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

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

Column 2 is A078039(n-2).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: [order 15]
k=4: [order 52] for n>54

A300646 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4 or 5 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 4, 2, 4, 0, 0, 9, 10, 10, 9, 0, 0, 19, 30, 64, 30, 19, 0, 0, 41, 104, 316, 316, 104, 41, 0, 0, 88, 340, 1789, 2998, 1789, 340, 88, 0, 0, 189, 1144, 9738, 28089, 28089, 9738, 1144, 189, 0, 0, 406, 3795, 54293, 268615, 456649, 268615, 54293, 3795

Offset: 1

R. H. Hardin, Mar 10 2018

Table starts
.0...0....0......0........0..........0............0..............0
.0...1....2......4........9.........19...........41.............88
.0...2....2.....10.......30........104..........340...........1144
.0...4...10.....64......316.......1789.........9738..........54293
.0...9...30....316.....2998......28089.......268615........2552014
.0..19..104...1789....28089.....456649......7455411......121708555
.0..41..340...9738...268615....7455411....209156119.....5850560104
.0..88.1144..54293..2552014..121708555...5850560104...280782187131
.0.189.3795.300456.24338903.1990954802.164070075524.13506744672543

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

R. H. Hardin, Table of n, a(n) for n = 1..241

Column 2 is A078039(n-2).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: [order 15]
k=4: [order 49]

A300776 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4 or 6 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 4, 3, 4, 0, 0, 9, 11, 11, 9, 0, 0, 19, 32, 36, 32, 19, 0, 0, 41, 102, 179, 179, 102, 41, 0, 0, 88, 330, 735, 1536, 735, 330, 88, 0, 0, 189, 1059, 3482, 10374, 10374, 3482, 1059, 189, 0, 0, 406, 3418, 15986, 78932, 116954, 78932, 15986, 3418

Offset: 1

R. H. Hardin, Mar 12 2018

Table starts
.0...0....0.....0.......0.........0...........0............0..............0
.0...1....2.....4.......9........19..........41...........88............189
.0...2....3....11......32.......102.........330.........1059...........3418
.0...4...11....36.....179.......735........3482........15986..........74347
.0...9...32...179....1536.....10374.......78932.......592064........4449947
.0..19..102...735...10374....116954.....1413931.....17336190......210669938
.0..41..330..3482...78932...1413931....27354464....538564737....10457318118
.0..88.1059.15986..592064..17336190...538564737..16992197390...534677180915
.0.189.3418.74347.4449947.210669938.10457318118.534677180915.27071152148341

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

R. H. Hardin, Table of n, a(n) for n = 1..219

Column 2 is A078039(n-2).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: [order 15]
k=4: [order 56] for n>57

A301400 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4, 5 or 6 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 4, 3, 4, 0, 0, 9, 13, 13, 9, 0, 0, 19, 44, 78, 44, 19, 0, 0, 41, 156, 446, 446, 156, 41, 0, 0, 88, 554, 2619, 4857, 2619, 554, 88, 0, 0, 189, 1963, 15538, 49325, 49325, 15538, 1963, 189, 0, 0, 406, 6964, 92338, 521514, 885003, 521514

Offset: 1

R. H. Hardin, Mar 20 2018

Table starts
.0...0....0......0........0..........0............0..............0
.0...1....2......4........9.........19...........41.............88
.0...2....3.....13.......44........156..........554...........1963
.0...4...13.....78......446.......2619........15538..........92338
.0...9...44....446.....4857......49325.......521514........5457161
.0..19..156...2619....49325.....885003.....16325745......299552320
.0..41..554..15538...521514...16325745....528025038....16971131642
.0..88.1963..92338..5457161..299552320..16971131642...956047339302
.0.189.6964.549096.57311190.5507220522.546618326745.53967719265287

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

R. H. Hardin, Table of n, a(n) for n = 1..263

Column 2 is A078039(n-2).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: [order 15]
k=4: [order 45]

A099509 Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 7, 1, 1, 5, 10, 16, 6, 3, 1, 6, 15, 30, 19, 16, 1, 1, 7, 21, 50, 45, 51, 10, 4, 1, 8, 28, 77, 90, 126, 45, 30, 1, 1, 9, 36, 112, 161, 266, 141, 126, 15, 5, 1, 10, 45, 156, 266, 504, 357, 393, 90, 50, 1, 1, 11, 55, 210, 414, 882, 784, 1016, 357

Offset: 0

Paul D. Hanna, Oct 20 2004

Row sums form absolute values of A078039. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).

			Rows begin:
[1],
[1,1],
[1,2,1],
[1,3,3,2],
[1,4,6,7,1],
[1,5,10,16,6,3],
[1,6,15,30,19,16,1],
[1,7,21,50,45,51,10,4],
[1,8,28,77,90,126,45,30,1],
[1,9,36,112,161,266,141,126,15,5],...
and can be derived from coefficients of (1+z+z^2)^n:
[1],
[1,1,1],
[1,2,3,2,1],
[1,3,6,7,6,3,1],
[1,4,10,16,19,16,10,4,1],
[1,5,15,30,45,51,45,30,15,5,1],...
by shifting each column k down by [k/2] rows.

Cf. A027907, A078039, A099510.

PARI
```
T(n,k)=if(n
				
```

G.f.: (1-x+x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2+x^3*y^2+x^4*y^4).

cp's OEIS Frontend

A002478 Bisection of A000930.

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A297224 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 1 neighboring 1s.

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A141015 a(0) = 0, a(1) = 1, a(2) = 2; for n > 2, a(n) = a(n-1) + 2*a(n-2) + a(n-3).

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A228482 T(n,k)=Number of nXk binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

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A280069 T(n,k)=Number of nXk 0..1 arrays with no element unequal to a strict majority of its horizontal, vertical and antidiagonal neighbors and with new values introduced in order 0 sequentially upwards.

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A300465 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3 or 4 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

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A300646 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4 or 5 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

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A300776 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4 or 6 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

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