A000400 -id:A000400 - cp's OEIS frontend

A224353 T(n,k)=Number of nXk 0..2 arrays with diagonals and antidiagonals unimodal and rows nondecreasing.

3, 6, 9, 10, 36, 27, 15, 100, 216, 81, 21, 225, 788, 1296, 243, 28, 441, 2321, 5880, 7776, 729, 36, 784, 5840, 19608, 45064, 46656, 2187, 45, 1296, 13052, 57387, 160362, 349280, 279936, 6561, 55, 2025, 26610, 151010, 495985, 1351748, 2710892, 1679616

Offset: 1

R. H. Hardin Apr 04 2013

Table starts
.....3........6.........10.........15..........21..........28...........36
.....9.......36........100........225.........441.........784.........1296
....27......216........788.......2321........5840.......13052........26610
....81.....1296.......5880......19608.......57387......151010.......363392
...243.....7776......45064.....160362......495985.....1421762......3816783
...729....46656.....349280....1351748.....4231138....12340932.....34697869
..2187...279936....2710892...11704964....37433596...107694133....300892325
..6561..1679616...21021916..102319662...342170839...977742699...2654062881
.19683.10077696..163012744..895494806..3178789749..9202126546..24422915139
.59049.60466176.1264202660.7833508842.29672959682.88363107023.233364588801

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

R. H. Hardin, Table of n, a(n) for n = 1..264

Column 1 is A000244
Column 2 is A000400
Row 1 is A000217(n+1)
Row 2 is A000537(n+1)

Empirical: columns k=1..6 have recurrences of order 1,1,10,26,56,98

Empirical: rows n=1..7 are polynomials of degree 2*n for k>0,0,1,3,5,7,9

A275228 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,1) or (-1,0) and new values introduced in order 0..2.

Original entry on oeis.org

1, 2, 1, 5, 6, 2, 14, 36, 11, 4, 41, 216, 61, 27, 8, 122, 1296, 339, 187, 66, 16, 365, 7776, 1885, 1302, 648, 162, 32, 1094, 46656, 10483, 9075, 6448, 2282, 404, 64, 3281, 279936, 58301, 63267, 64248, 32388, 8134, 1007, 128, 9842, 1679616, 324243, 441090, 640250

Offset: 1

R. H. Hardin, Jul 20 2016

Table starts
...1....2......5.......14.........41.........122...........365............1094
...1....6.....36......216.......1296........7776.........46656..........279936
...2...11.....61......339.......1885.......10483.........58301..........324243
...4...27....187.....1302.......9075.......63267........441090.........3075255
...8...66....648.....6448......64248......640250.......6380362........63583084
..16..162...2282....32388.....460034.....6534900......92831330......1318716132
..32..404...8134...164645....3334854....67550874....1368326231.....27717164160
..64.1007..29027...844176...24595019...716735348...20887845278....608741311725
.128.2512.103456..4319300..181187962..7605814181..319348056936..13409199459965
.256.6271.368889.22083548.1332436071.80496567059.4865299894416.294093896351121

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

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

Column 1 is A000079(n-2).
Row 1 is A007051(n-1).
Row 2 is A000400(n-1).
Row 4 is A078100.

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>2
k=2: a(n) = 3*a(n-1) -a(n-2) -2*a(n-4) +a(n-5) for n>6
k=3: [order 9] for n>13
k=4: [order 31] for n>35
k=5: [order 81] for n>86
Empirical for row n:
n=1: a(n) = 4*a(n-1) -3*a(n-2)
n=2: a(n) = 6*a(n-1)
n=3: a(n) = 7*a(n-1) -8*a(n-2)
n=4: a(n) = 9*a(n-1) -15*a(n-2) +6*a(n-3)
n=5: a(n) = 11*a(n-1) -9*a(n-2) -15*a(n-3) +20*a(n-4) -6*a(n-5) for n>6
n=6: a(n) = 19*a(n-1) -75*a(n-2) +101*a(n-3) -44*a(n-4) for n>5
n=7: a(n) = 20*a(n-1) +57*a(n-2) -1206*a(n-3) +3096*a(n-4) +2306*a(n-5) -16957*a(n-6) +20440*a(n-7) -7755*a(n-8) for n>9

A009980 Powers of 36.

Original entry on oeis.org

1, 36, 1296, 46656, 1679616, 60466176, 2176782336, 78364164096, 2821109907456, 101559956668416, 3656158440062976, 131621703842267136, 4738381338321616896, 170581728179578208256, 6140942214464815497216, 221073919720733357899776, 7958661109946400884391936

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 36), L(1, 36), P(1, 36), T(1, 36). Essentially same as Pisot sequences E(36, 1296), L(36, 1296), P(36, 1296), T(36, 1296). See A008776 for definitions of Pisot sequences.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 36-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
See David Applegate's comment in A000244 from Feb 20 2017 for a proof of Janjic's assertion. - Alonso del Arte, Sep 03 2017

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (36).

Cf. A000079, A000244, A000400, A001027, A005843, A008776.

[36^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

36^Range[0, 20]  (* Harvey P. Dale, Mar 04 2011 *)

a(n)=36^n \\ Charles R Greathouse IV, Nov 18 2011

G.f.: 1/(1-36*x). - Philippe Deléham, Nov 24 2008
a(n) = 36^n; a(n) = 36*a(n-1) for n > 0, a(0) = 1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(36*x).
a(n) = A000079(n)*A001027(n) = A000400(A005843(n)). (End)

A038255 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

Original entry on oeis.org

1, 6, 1, 36, 12, 1, 216, 108, 18, 1, 1296, 864, 216, 24, 1, 7776, 6480, 2160, 360, 30, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 279936, 326592, 163296, 45360, 7560, 756, 42, 1, 1679616, 2239488, 1306368, 435456, 90720, 12096, 1008

Offset: 0

N. J. A. Sloane

T(n,k) = A013613(n,n-k), 0 <= k <= n. - Reinhard Zumkeller, Nov 21 2013

			1
6, 1
36, 12, 1
216, 108, 18, 1
1296, 864, 216, 24, 1
7776, 6480, 2160, 360, 30, 1
46656, 46656, 19440, 4320, 540, 36, 1
279936, 326592, 163296, 45360, 7560, 756, 42, 1
1679616, 2239488, 1306368, 435456, 90720, 12096, 1008, 48, 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A038207.

Cf. A000420 (row sums), A013613 (mirrored), A110440, A007318, A000400.

a038255 n k = a038255_tabl !! n !! k
a038255_row n = a038255_tabl !! n
a038255_tabl = map reverse a013613_tabl
-- Reinhard Zumkeller, Nov 21 2013

for i from 0 to 8 do seq(binomial(i, j)*6^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

Table[Binomial[n,m]6^(n-m),{n,0,10},{m,0,n}]//Flatten (* Harvey P. Dale, Dec 25 2019 *)

G.f.: 1/(1 - 6*x - x*y). - Ilya Gutkovskiy, Apr 21 2017

A212700 a(n) = 5n6^(n-1).

Original entry on oeis.org

5, 60, 540, 4320, 32400, 233280, 1632960, 11197440, 75582720, 503884800, 3325639680, 21767823360, 141490851840, 914248581120, 5877312307200, 37614798766080, 239794342133760, 1523399350026240, 9648195883499520, 60935974001049600, 383896636206612480, 2413064570441564160

Offset: 1

Stanislav Sykora, May 25 2012

Main transitions in systems of n particles with spin 5/2.
Refer to the general explanation in A212697.
This particular sequence is obtained for base b=6, corresponding to spin S=(b-1)/2=5/2.
Arithmetic derivative of 6^n: a(n) = A003415(6^n). - Bruno Berselli, Oct 22 2013

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A001787, A212697, A212698, A212699, A212701, A212702, A212703, A212704 (b = 2, 3, 4, 5, 7, 8, 9, 10).

Cf. A000400, A003415, A008587, A053469.

Rest@ CoefficientList[Series[5 x/(6 x - 1)^2, {x, 0, 18}], x] (* or *)
Array[5 # 6^(# - 1) &, 18] (* Michael De Vlieger, Nov 18 2019 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212700.txt", n, " ", mtrans(n, 6)))

a(n) = n*(b-1)*b^(n-1): for this sequence, set b=6.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 5*x/(6*x-1)^2.
a(n) = 5*A053469(n). (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 5*x*exp(6*x).
a(n) = A008587(n)*A000400(n-1).
a(n) = 12*a(n-1) - 36*a(n-2) for n > 2. (End)

A275142 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-2) (-1,-2) or (0,-1) and new values introduced in order 0..2.

Original entry on oeis.org

1, 1, 2, 2, 6, 5, 4, 16, 36, 14, 8, 48, 80, 216, 41, 16, 144, 224, 400, 1296, 122, 32, 432, 528, 1088, 2000, 7776, 365, 64, 1296, 1216, 2320, 5248, 10000, 46656, 1094, 128, 3888, 2816, 6464, 9744, 25344, 50000, 279936, 3281, 256, 11664, 6544, 17872, 32384, 41360

Offset: 1

R. H. Hardin, Jul 17 2016

Table starts
....1........1.......2........4........8........16........32.........64
....2........6......16.......48......144.......432......1296.......3888
....5.......36......80......224......528......1216......2816.......6544
...14......216.....400.....1088.....2320......6464.....17872......49792
...41.....1296....2000.....5248.....9744.....32384....107472.....362176
..122.....7776...10000....25344....41360....165568....663904....2695808
..365....46656...50000...122368...175120....841536...4055152...19906560
.1094...279936..250000...590848...741904...4283968..24875600..147762240
.3281..1679616.1250000..2852864..3142672..21800000.152379136.1093999424
.9842.10077696.6250000.13774848.13312656.110943552.933805200.8109111360

			Some solutions for n=5 k=4
..0..1..2..1. .0..1..0..1. .0..1..0..2. .0..1..2..1. .0..1..2..0
..2..0..1..0. .0..2..1..2. .2..0..1..2. .2..0..2..0. .0..1..2..0
..2..0..1..2. .1..2..1..0. .1..2..1..2. .2..0..1..2. .2..0..2..0
..1..2..0..1. .0..1..2..1. .0..2..0..1. .2..0..1..2. .2..0..1..2
..0..2..0..1. .2..1..2..0. .0..1..2..0. .1..2..0..2. .0..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..721

Column 1 is A007051(n-1).
Column 2 is A000400(n-1).
Column 3 is A055842.
Row 1 is A000079(n-2).

Empirical for column k:
k=1: a(n) = 4*a(n-1) -3*a(n-2)
k=2: a(n) = 6*a(n-1)
k=3: a(n) = 5*a(n-1) for n>2
k=4: a(n) = 4*a(n-1) +4*a(n-2) for n>3
k=5: a(n) = 3*a(n-1) +5*a(n-2) +a(n-3) for n>4
k=6: a(n) = 3*a(n-1) +10*a(n-2) +4*a(n-3) -4*a(n-4) for n>6
k=7: a(n) = 3*a(n-1) +18*a(n-2) +11*a(n-3) -23*a(n-4) -4*a(n-5) for n>7
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 3*a(n-1) for n>3
n=3: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3) for n>5
n=4: a(n) = 5*a(n-1) -9*a(n-2) +10*a(n-3) -6*a(n-4) +a(n-5) for n>9
n=5: [order 8] for n>12
n=6: [order 13] for n>18
n=7: [order 21] for n>27

A089504 A generalization of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, 6, 1, 36, 30, 1, 216, 756, 90, 1, 1296, 18360, 6156, 210, 1, 7776, 441936, 387720, 31356, 420, 1, 46656, 10614240, 23705136, 4150440, 119556, 756, 1, 279936, 254788416, 1432922400, 521757936, 29257200, 373572, 1260, 1, 1679616

Offset: 1

Wolfdieter Lang, Dec 01 2003

This triangle underlies the array entry A078741 ((3,3)-generalized Stirling2).

For the computation of the column sequences see A089505.

			[1]; [6,1]; [36,30,1]; [216,756,90,1]; ...
a(3,2) = 30 = ((-1)*(3*2*1)^1 + 4*(4*3*2)^1)/3.

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
W. Lang, First 8 rows.

Cf. A071951 (Legendre-Stirling, (2, 2) case).
The column sequences (without leading zeros) are A000400 (powers of 6), A089507, A089513-4, etc.
Cf. A008279, A089505, A089506.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 2, 3]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max - m + 1), x]; a[n_, m_] := col[m][[n - m + 1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

G.f. for m-th column sequence (without leading zeros and m>=1) is 1/Product_{r=1..m} 1-fallfac(r+2, 3)*x with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = Sum_{p=1..m} A089505(m, p)*((p+2)*(p+1)*p)^(n-m))/D(m) if n>=m>=1 else 0; with D(m) := A089506(m).

A090018 a(n) = 6a(n-1) + 3a(n-2) for n > 2, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 39, 252, 1629, 10530, 68067, 439992, 2844153, 18384894, 118841823, 768205620, 4965759189, 32099171994, 207492309531, 1341251373168, 8669985167601, 56043665125110, 362271946253463, 2341762672896108, 15137391876137037, 97849639275510546, 632510011281474387

Offset: 0

Paul Barry, Nov 19 2003

From Johannes W. Meijer, Aug 09 2010: (Start)

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner or side square on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032. The central square leads to A180028. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,3).

Cf. A141041, A180028, A180032.

Sequences with g.f. of the form 1/(1 - 6*x - k*x^2): A106392 (k=-10), A027471 (k=-9), A006516 (k=-8), A081179 (k=-7), A030192 (k=-6), A003463 (k=-5), A084326 (k=-4), A138395 (k=-3), A154244 (k=-2), A001109 (k=-1), A000400 (k=0), A005668 (k=1), A135030 (k=2), this sequence (k=3), A135032 (k=4), A015551 (k=5), A057089 (k=6), A015552 (k=7), A189800 (k=8), A189801 (k=9), A190005 (k=10), A015553 (k=11).

[n le 2 select 6^(n-1) else 6*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

a:= n-> (<<0|1>, <3|6>>^n. <<1,6>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 17 2011

Join[{a=1,b=6},Table[c=6*b+3*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,3}, {1,6}, 41] (* G. C. Greubel, Oct 10 2022 *)

my(x='x+O('x^30)); Vec(1/(1-6*x-3*x^2)) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,6,-3) for n in range(1, 31)] # Zerinvary Lajos, Apr 24 2009

a(n) = (3+2*sqrt(3))^n*(sqrt(3)/4+1/2) + (1/2-sqrt(3)/4)*(3-2*sqrt(3))^n.
a(n) = (-i*sqrt(3))^n * ChebyshevU(n, isqrt(3)), i^2=-1.
From Johannes W. Meijer, Aug 09 2010: (Start)
G.f.: 1/(1 - 6*x - 3*x^2).
Limit_{k->oo} a(n+k)/a(k) = A141041(n) + A090018(n-1)*sqrt(12) for n >= 1.
Limit_{n->oo} A141041(n)/A090018(n-1) = sqrt(12). (End)
a(n) = Sum_{k=0..n} A099089(n,k)*3^k. - Philippe Deléham, Nov 21 2011
E.g.f.: exp(3*x)*(2*cosh(2*sqrt(3)*x) + sqrt(3)*sinh(2*sqrt(3)*x))/2. - Stefano Spezia, Apr 23 2025

Typo in Mathematica program corrected by Vincenzo Librandi, Nov 15 2011

A167747 a(n) = phi(6^n).

Original entry on oeis.org

1, 2, 12, 72, 432, 2592, 15552, 93312, 559872, 3359232, 20155392, 120932352, 725594112, 4353564672, 26121388032, 156728328192, 940369969152, 5642219814912, 33853318889472, 203119913336832, 1218719480020992, 7312316880125952, 43873901280755712, 263243407684534272

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Bevan, D. Levin, P. Nugent, J. Pantone, L. Pudwell, M. Riehl and M. L. Tlachac, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A000010, A000400, A132022.

Mathematica
```
Table[EulerPhi[6^n],{n,0,40}]
```

a(n) = eulerphi(6^n); \\ Michel Marcus, Jan 02 2021

a(n+1) = 2*6^n. - Charles R Greathouse IV, Nov 12 2009
G.f.: (1-4x)/(1-6x). - Philippe Deléham, Oct 10 2011
a(n) = ((8*n-4)*a(n-1)-12*(n-2)*a(n-2))/n, a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 19 2012
Sum_{n>=0} 1/a(n) = 8/5. - Amiram Eldar, Jan 02 2021
a(n) = A000010(A000400(n)). - Michel Marcus, Jan 02 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} (-1)^n/a(n) = 4/7.
Product_{n>=1} (1 - 1/a(n)) = A132022. (End)

A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 6x - 3*x^2).

Original entry on oeis.org

1, 9, 57, 369, 2385, 15417, 99657, 644193, 4164129, 26917353, 173996505, 1124731089, 7270376049, 46996449561, 303789825513, 1963728301761, 12693739287105, 82053620627913, 530402941628793, 3428578511656497

Offset: 0

Johannes W. Meijer, Aug 09 2010; edited Jun 21 2013

The a(n) represent the number of n-move routes of a fairy chess piece starting in the center square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen.
On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the center square (off the center square the piece behaves like a normal queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the center square the 512 red queens lead to 17 red queen sequences, see the overview of red queen sequences and the crossreferences.
The sequence above corresponds to just one red queen vector, i.e., A[5] = [111 111 111] vector. The other squares lead for this vector to A090018.
This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 6*x - k*x^2). The members of this family that are red queen sequences are A180028 (k=3; this sequence), A180029 (k=2), A015451 (k=1), A000400 (k=0), A001653 (k=-1), A180034 (k=-2), A084120 (k=-3), A154626 (k=-4) and A000012 (k=-5). Other members of this family are A123362 (k=5), 6*A030192(k=-6).
Inverse binomial transform of A107903.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Johannes W. Meijer, The red queen sequences.
Wikipedia, Alice in Wonderland (2010 film).
Index entries for linear recurrences with constant coefficients, signature (6, 3).

Cf. A180140 (berserker sequences)
Cf. A180032 (Corner and side squares).
Cf. Red queen sequences center square [decimal value A[5]]: A180028 [511], A180029 [255], A180031 [495], A015451 [127], A152240 [239], A000400 [63], A057088 [47], A001653 [31], A122690 [15], A180034 [23], A180036 [7], A084120 [19], A180038 [3], A154626 [17], A015449 [1], A000012 [16], A000007 [0].

I:=[1,9]; [n le 2 select I[n] else 6*Self(n-1)+3*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 15 2011

nmax:=19; m:=5; A[1]:=[0,1,1,1,1,0,1,0,1]: A[2]:=[1,0,1,1,1,1,0,1,0]: A[3]:=[1,1,0,0,1,1,1,0,1]: A[4]:=[1,1,0,0,1,1,1,1,0]: A[5]:=[1,1,1,1,1,1,1,1,1]: A[6]:=[0,1,1,1,1,0,0,1,1]: A[7]:=[1,0,1,1,1,0,0,1,1]: A[8]:=[0,1,0,1,1,1,1,0,1]: A[9]:=[1,0,1,0,1,1,1,1,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{6,3},{1,9},50] (* Vincenzo Librandi, Nov 15 2011 *)

G.f.: (1+3*x)/(1 - 6*x - 3*x^2).
a(n) = 6*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 9.
a(n) = ((1-A)*A^(-n-1) + (1-B)*B^(-n-1))/4 with A=(-1+2*sqrt(3)/3) and B=(-1-2*sqrt(3)/3).
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n-1)*A108411(n+1)/(A041017(n-1)*sqrt(12) - A041016(n-1)) for n >= 1.

cp's OEIS Frontend

A224353 T(n,k)=Number of nXk 0..2 arrays with diagonals and antidiagonals unimodal and rows nondecreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A275228 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,1) or (-1,0) and new values introduced in order 0..2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A009980 Powers of 36.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A038255 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A212700 a(n) = 5*n*6^(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A275142 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-2) (-1,-2) or (0,-1) and new values introduced in order 0..2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A089504 A generalization of triangle A071951 (Legendre-Stirling).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A090018 a(n) = 6*a(n-1) + 3*a(n-2) for n > 2, a(0)=1, a(1)=6.

Views

Author

Keywords

Comments

A212700 a(n) = 5n6^(n-1).

A090018 a(n) = 6a(n-1) + 3a(n-2) for n > 2, a(0)=1, a(1)=6.

A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 6x - 3*x^2).