A000400 -id:A000400 - cp's OEIS frontend

A223357 T(n,k)=Rolling cube face footprints: number of nXk 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal or antidiagonal neighbor moves across a corresponding cube edge.

1, 4, 6, 16, 64, 36, 64, 768, 1024, 216, 256, 9216, 36864, 16384, 1296, 1024, 110592, 1327104, 1769472, 262144, 7776, 4096, 1327104, 48365568, 191102976, 84934656, 4194304, 46656, 16384, 15925248, 1764753408, 21177040896, 27518828544

Offset: 1

R. H. Hardin Mar 19 2013

Table starts
....1.......4.........16............64..............256................1024
....6......64........768..........9216...........110592.............1327104
...36....1024......36864.......1327104.........48365568..........1764753408
..216...16384....1769472.....191102976......21177040896.......2356125106176
.1296..262144...84934656...27518828544....9273505480704....3147420753985536
.7776.4194304.4076863488.3962711310336.4060947412942848.4204783428144463872

			Some solutions for n=3 k=4
..0..2..0..2....0..3..5..2....0..2..5..4....0..4..0..4....0..1..3..4
..0..4..0..1....0..3..5..3....0..4..0..1....0..4..0..4....0..1..0..1
..0..4..5..4....0..1..0..1....0..4..2..1....0..2..0..1....0..1..2..4

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

Column 1 is A000400(n-1)
Column 2 is A013709 (n-1)
Column 3 is 16*48^(n-1)
Column 4 is 64*144^(n-1)
Row 1 is A000302(n-1)
Row 2 is 64*12^(n-2) for n>1

Empirical for column k:
k=1: a(n) = 6*a(n-1)
k=2: a(n) = 16*a(n-1)
k=3: a(n) = 48*a(n-1)
k=4: a(n) = 144*a(n-1)
k=5: a(n) = 480*a(n-1) -18432*a(n-2)
k=6: a(n) = 1600*a(n-1) -368640*a(n-2) +21233664*a(n-3)
k=7: a(n) = 5376*a(n-1) -5750784*a(n-2) +2038431744*a(n-3) -217432719360*a(n-4)
Empirical for row n:
n=1: a(n) = 4*a(n-1)
n=2: a(n) = 12*a(n-1) for n>2
n=3: a(n) = 40*a(n-1) -128*a(n-2) for n>4
n=4: a(n) = 144*a(n-1) -3840*a(n-2) +24576*a(n-3) for n>7
n=5: [order 7] for n>11
n=6: [order 9] for n>15
n=7: [order 27] for n>33

A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 36, 8, 1, 1, 1, 120, 576, 216, 16, 1, 1, 1, 720, 14400, 13824, 1296, 32, 1, 1, 1, 5040, 518400, 1728000, 331776, 7776, 64, 1, 1, 1, 40320, 25401600, 373248000, 207360000, 7962624, 46656, 128, 1, 1

Offset: 0

Alois P. Heinz, Jul 29 2013

A(n,k) is the determinant of the k X k matrix M = [Stirling2(n+i,j)] for 1<=i,j<=k.  A(2,3) = det([1,3,1; 1,7,6; 1,15,25]) = 36.
A(n,k) is the determinant of the symmetric k X k matrix M = [sigma_n(gcd(i,j))] for 1<=i,j<=k.  A(2,3) = det([1,1,1; 1,5,1; 1,1,10]) = 36.
A(n,k) is (-1)^(n*k) times the determinant of the n X n matrix M = [Stirling1(k+i,j)] for 1<=i,j<=n.  A(2,3) = (-1)^(2+3) * det([-6,11; 24,-50]) = 36.
A(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_i-p_j) <= 1 for 1<=i,j<=k.  A(2,3) = 36:
          (1,2,2)-(1,1,2)         (0,1,1)-(0,0,1)
         /       X       \       /       X       \
  (2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0).
         \       X       /       \       X       /
          (2,2,1) (2,1,1)         (1,1,0) (1,0,0)
A(n,k) is the number of set partitions of [k*(n+1)] into k blocks of size n+1 such that the elements of each block are distinct mod n+1.  A(2,3) = 36: 123|456|789, 126|345|789, ..., 189|234|567, 189|246|357.

			Square array A(n,k) begins:
  1, 1,  1,    1,       1,           1, ...
  1, 1,  2,    6,      24,         120, ...
  1, 1,  4,   36,     576,       14400, ...
  1, 1,  8,  216,   13824,     1728000, ...
  1, 1, 16, 1296,  331776,   207360000, ...
  1, 1, 32, 7776, 7962624, 24883200000, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns k=0+1, 2-4 give: A000012, A000079, A000400, A009968.
Rows n=0-4 give: A000012, A000142, A001044, A000442, A134375.
Main diagonal gives: A036740.
Cf. A008275, A048994, A008277, A048993, A003989, A109004, A109974.

A:= (n, k)-> k!^n:
seq(seq(A(n,d-n), n=0..d), d=0..12);

A(n,k) = (k!)^n.
A(n,k) = k^n * A(n,k-1) for k>0, A(n,0) = 1.
A(n,k) = k!  * A(n-1,k) for n>0, A(0,k) = 1.
G.f. of column k: 1/(1-k!*x).

A233155 T(n,k) = Number of n X k 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally or antidiagonally.

Original entry on oeis.org

3, 6, 9, 12, 24, 27, 24, 72, 96, 81, 48, 216, 432, 384, 243, 96, 648, 1944, 2592, 1536, 729, 192, 1944, 8856, 17496, 15552, 6144, 2187, 384, 5832, 40392, 121176, 157464, 93312, 24576, 6561, 768, 17496, 184248, 842616, 1658232, 1417176, 559872, 98304, 19683

Offset: 1

R. H. Hardin, Dec 05 2013

Table starts
.....3.......6........12.........24...........48.............96
.....9......24........72........216..........648...........1944
....27......96.......432.......1944.........8856..........40392
....81.....384......2592......17496.......121176.........842616
...243....1536.....15552.....157464......1658232.......17587584
...729....6144.....93312....1417176.....22692312......367125912
..2187...24576....559872...12754584....310536504.....7663517136
..6561...98304...3359232..114791256...4249585944...159971190624
.19683..393216..20155392.1033121304..58154132088..3339300422232
.59049.1572864.120932352.9298091736.795819434328.69705848287656

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

R. H. Hardin, Table of n, a(n) for n = 1..390

Column 1 is A000244.
Column 2 is A002023(n-1).
Column 3 is 2*A000400.
Column 4 is 3*A055275.
Row 1 is A003945.
Row 2 is A005051(n-1) for n>1.

Empirical for column k:
k=1: a(n) = 3*a(n-1).
k=2: a(n) = 4*a(n-1).
k=3: a(n) = 6*a(n-1).
k=4: a(n) = 9*a(n-1).
k=5: a(n) = 15*a(n-1) -18*a(n-2).
k=6: a(n) = 25*a(n-1) -90*a(n-2) +81*a(n-3).
k=7: a(n) = 42*a(n-1) -351*a(n-2) +972*a(n-3) -810*a(n-4).
Empirical for row n:
n=1: a(n) = 2*a(n-1).
n=2: a(n) = 3*a(n-1) for n>2.
n=3: a(n) = 5*a(n-1) -2*a(n-2) for n>4.
n=4: a(n) = 9*a(n-1) -15*a(n-2) +6*a(n-3) for n>7.
n=5: [order 7] for n>11.
n=6: [order 9] for n>15.
n=7: [order 27] for n>33.

A240733 a(n) = floor(6^n/(2+2*cos(Pi/9))^n).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 32, 50, 78, 121, 187, 289, 448, 693, 1072, 1658, 2564, 3966, 6134, 9487, 14673, 22695, 35101, 54288, 83964, 129862, 200850, 310643, 480452, 743085, 1149282, 1777523, 2749182, 4251987, 6576279, 10171116, 15731022, 24330178, 37629950

Offset: 0

Kival Ngaokrajang, Apr 11 2014

a(n) is the perimeter (rounded down) of a nonaflake after n iterations, let a(0) = 1. The total number of sides is 9*A000400(n). The total number of holes is A002452(n). 2*cos(Pi/9) = 1.87938524... = diagonal b of nonagon (see comments in A123609).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of nonaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A002452, A123609, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).

A240733:=n->floor(6^n/(2+2*cos(Pi/9))^n); seq(A240733(n), n=0..50); # Wesley Ivan Hurt, Apr 12 2014

Table[Floor[6^n/(2 + 2*Cos[Pi/9])^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 12 2014 *)

{a(n)=floor(6^n/(2+2*cos(Pi/9))^n)}
       for (n=0, 100, print1(a(n), ", "))

A240734 a(n) = floor(6^n/(2+sqrt(5))^n).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 11, 16, 22, 32, 46, 65, 92, 130, 185, 262, 371, 526, 745, 1056, 1496, 2119, 3001, 4251, 6021, 8528, 12080, 17110, 24236, 34328, 48622, 68869, 97547, 138166, 195700, 277191, 392616, 556104, 787670, 1115663, 1580234, 2238256, 3170284

Offset: 0

Kival Ngaokrajang, Apr 11 2014

a(n) is the perimeter (rounded down) of a decaflake after n iterations, let a(0) = 1. The total number of sides is 10*A000400(n). The total number of holes is A002275(n). 2 + sqrt(5) = A098317.

Kival Ngaokrajang, Illustration of decaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A002275, A098317, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).

A240734:=n->floor(6^n/(2+sqrt(5))^n); seq(A240734(n), n=0..50); # Wesley Ivan Hurt, Apr 12 2014

Table[Floor[6^n/(2 + Sqrt[5])^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 12 2014 *)

{a(n)=floor(6^n/(2+sqrt(5))^n)}
       for (n=0, 100, print1(a(n), ", "))

A240735 a(n) = floor(6^n/(3+sqrt(3))^n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 13, 17, 21, 27, 35, 44, 56, 71, 90, 115, 146, 185, 235, 298, 378, 479, 607, 770, 977, 1238, 1570, 1991, 2525, 3202, 4060, 5148, 6527, 8276, 10494, 13306, 16872, 21393, 27125, 34393, 43609, 55294, 70111, 88897, 112717, 142919

Offset: 0

Kival Ngaokrajang, Apr 11 2014

a(n) is the perimeter (rounded down) of a dodecaflake after n iterations, let a(0) = 1. The total number of sides is 12*A000400(n). The total number of holes is A240846. 3 + sqrt(3) = A165663.

Kival Ngaokrajang, Illustration of dodecaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A240846, A165663, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).

A240735:=n->floor(6^n/(3+sqrt(3))^n); seq(A240735(n), n=0..50); # Wesley Ivan Hurt, Apr 12 2014

Table[Floor[6^n/(3 + Sqrt[3])^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 12 2014 *)

{a(n)=floor(6^n/(3+sqrt(3))^n)}
       for (n=0, 100, print1(a(n), ", "))

A339686 a(n) = Sum_{d|n} 6^(d-1).

Original entry on oeis.org

1, 7, 37, 223, 1297, 7819, 46657, 280159, 1679653, 10078999, 60466177, 362805091, 2176782337, 13060740679, 78364165429, 470185264735, 2821109907457, 16926661132171, 101559956668417, 609359750089711, 3656158440109669, 21936950700844039, 131621703842267137

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 6 of A308813.
Cf. A000400, A320071.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), this sequence (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A339686:= func< n | (&+[6^(d-1): d in Divisors(n)]) >;
[A339686(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[6^(d - 1), {d, Divisors[n]}], {n, 1, 23}]
nmax = 23; CoefficientList[Series[Sum[x^k/(1 - 6 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 6^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339686(n): return sum(6^(k-1) for k in (1..n) if (k).divides(n))
[A339686(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 6*x^k).
G.f.: Sum_{k>=1} 6^(k-1) * x^k / (1 - x^k).
a(n) ~ 6^(n-1). - Vaclav Kotesovec, Jun 05 2021

A056452 a(n) = 6^floor((n+1)/2).

Original entry on oeis.org

1, 6, 6, 36, 36, 216, 216, 1296, 1296, 7776, 7776, 46656, 46656, 279936, 279936, 1679616, 1679616, 10077696, 10077696, 60466176, 60466176, 362797056, 362797056, 2176782336, 2176782336, 13060694016, 13060694016, 78364164096

Offset: 0

Marks R. Nester

Number of achiral rows of length n using up to six different colors. For a(3) = 36, the rows are AAA, ABA, ACA, ADA, AEA, AFA, BAB, BBB, BCB, BDB, BEB, BFB, CAC, CBC, CCC, CDC, CEC, CFC, DAD, DBD, DCD, DDD, DED, DFD, EAE, EBE, ECE, EDE, EEE, EFE, FAF, FBF, FCF, FDF, FEF, and FFF. - Robert A. Russell, Nov 08 2018

Also: a(n) is the number of palindromes with n digits using a maximum of six different symbols. - David A. Corneth, Nov 09 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Column k=6 of A321391.
Cf. A016116.
Cf. A000400 (oriented), A056308 (unoriented), A320524 (chiral).

[6^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Maple
```
A056452:=n->6^floor((n+1)/2);
```

Riffle[6^Range[0, 20], 6^Range[20]] (* Harvey P. Dale, Jun 18 2017 *)
Table[6^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 6}, {1, 6}, 40] (* Robert A. Russell, Nov 08 2018 *)

a(n) = 6^floor((n+1)/2).
a(n) = 6*a(n-2). - Colin Barker, May 06 2012
G.f.: (1+6*x) / (1-6*x^2). - Colin Barker, May 06 2012 [Adapted to offset 0 by Robert A. Russell, Nov 08 2018]
a(n) = C(6,0)*A000007(n) + C(6,1)*A057427(n) + C(6,2)*A056453(n) + C(6,3)*A056454(n) + C(6,4)*A056455(n) + C(6,5)*A056456(n) + C(6,6)*A056457(n). - Robert A. Russell, Nov 08 2018

a(0)=1 prepended by Robert A. Russell, Nov 08 2018

Name corrected by David A. Corneth, Nov 08 2018

A075501 Stirling2 triangle with scaled diagonals (powers of 6).

Original entry on oeis.org

1, 6, 1, 36, 18, 1, 216, 252, 36, 1, 1296, 3240, 900, 60, 1, 7776, 40176, 19440, 2340, 90, 1, 46656, 489888, 390096, 75600, 5040, 126, 1, 279936, 5925312, 7511616, 2204496, 226800, 9576, 168, 1, 1679616, 71383680

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(6*z) - 1)*x/6) - 1.

			[1]; [6,1]; [36,18,1]; ...; p(3,x) = x(36 + 18*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      6       1
*     36      18       1
*    216     252      36       1
*   1296    3240     900      60      1
*   7776   40176   19440    2340     90    1
*  46656  489888  390096   75600   5040  126   1
* 279936 5925312 7511616 2204496 226800 9576 168 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000400, A016175, A075916-A075920. Row sums are A005012.

Cf. A075500, A075502.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 6^n, 9); # Peter Luschny, Jan 28 2016

Flatten[Table[6^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[6^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(6^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (6^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*6)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 6m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-6k*x), m >= 1.
E.g.f. for m-th column: (((exp(6x)-1)/6)^m)/m!, m >= 1.

A128964 a(n) = (n^3-n)*6^n.

Original entry on oeis.org

0, 216, 5184, 77760, 933120, 9797760, 94058496, 846526464, 7255941120, 59861514240, 478892113920, 3735358488576, 28524555730944, 213934167982080, 1579821548175360, 11510128422420480, 82872924641427456, 590469588070170624, 4168020621671792640, 29176144351702548480

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (24,-216,864,-1296).

Cf. A000400, A007531, A036289, A081144, A128796.

Cf. A128960, A128961, A128962, A128963, A128965, A128967, A128969.

[(n^3-n)*6^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

I:=[0, 216, 5184, 77760]; [n le 4 select I[n] else 24*Self(n-1) -216*Self(n-2) +864*Self(n-3) -1296*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

CoefficientList[Series[216 x/(1 - 6 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 216*x^2/(1-6*x)^4.
a(n) = 216*A081144(n+1). (End)
a(n) = 24*a(n-1) - 216*a(n-2) + 864*a(n-3) - 1296*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=2} 1/a(n) = 25*log(6/5)/12 - 3/8.
Sum_{n>=2} (-1)^n/a(n) = 49*log(7/6)/12 - 5/8. (End)
a(n) = A007531(n+1)*A000400(n). - Amiram Eldar, Oct 02 2022

Corrected offset. - Mohammad K. Azarian, Nov 20 2008

cp's OEIS Frontend

A223357 T(n,k)=Rolling cube face footprints: number of nXk 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal or antidiagonal neighbor moves across a corresponding cube edge.

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A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

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A233155 T(n,k) = Number of n X k 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally or antidiagonally.

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A240733 a(n) = floor(6^n/(2+2*cos(Pi/9))^n).

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A240734 a(n) = floor(6^n/(2+sqrt(5))^n).

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A240735 a(n) = floor(6^n/(3+sqrt(3))^n).

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Maple

Mathematica

PARI

A339686 a(n) = Sum_{d|n} 6^(d-1).

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Magma

Mathematica

PARI

SageMath

Formula

A056452 a(n) = 6^floor((n+1)/2).

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