A000312 -id:A000312 - cp's OEIS frontend

A081721 Number of bracelets of n beads in up to n colors.

1, 3, 10, 55, 377, 4291, 60028, 1058058, 21552969, 500280022, 12969598086, 371514016094, 11649073935505, 396857785692525, 14596464294191704, 576460770691256356, 24330595997127372497, 1092955780817066765469, 52063675152021153895330, 2621440000054016000176044

Offset: 1

N. J. A. Sloane, based on information supplied by Gary W. Adamson, Apr 05 2003

nonn

T(n,n), T given in A081720.
From Olivier Gérard, Aug 01 2016: (Start)
Number of classes of functions of [n] to [n] under rotation and reversal.
.
Classes can be of size between 1 and 2n
depending on divisibility properties of n.
.
n   1   2   3   4    5         n     2n
----------------------------------------
1   1
2   2   1
3   3   0   6                         1
4   4   6   0                 30     15
5   5   0   0                120    252
6   6  15  30                725   3515
7   7   0   0               2394  57627
.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000312 All endofunctions
Cf. A000169 Classes under translation mod n
Cf. A001700 Classes under sort
Cf. A056665 Classes under rotation
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
Cf. A275549 Classes under reversal
Cf. A275550 Classes under reversal and complement
Cf. A275551 Classes under translation and reversal
Cf. A275552 Classes under translation and complement
Cf. A275553 Classes under translation, complement and reversal
Cf. A275554 Classes under translation, rotation and complement
Cf. A275555 Classes under translation, rotation and reversal
Cf. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal
Row sums of partition array A213941.
Main diagonal of A321791.

Table[CycleIndex[DihedralGroup[n],s]/.Table[s[i]->n,{i,1,n}],{n,1,20}] (* Geoffrey Critzer, Jun 18 2013 *)
t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]); a[n_] := t[n, n]; Array[a, 20] (* Jean-François Alcover, Nov 02 2017, after Maple code for A081720 *)

a(n) ~ n^(n-1) / 2. - Vaclav Kotesovec, Mar 18 2017

Name changed by Olivier Gérard, Aug 05 2016

Name revised by Álvar Ibeas, Apr 20 2018

A343656 Array read by antidiagonals where A(n,k) is the number of divisors of n^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 5, 2, 1, 1, 6, 5, 7, 3, 4, 1, 1, 7, 6, 9, 4, 9, 2, 1, 1, 8, 7, 11, 5, 16, 3, 4, 1, 1, 9, 8, 13, 6, 25, 4, 7, 3, 1, 1, 10, 9, 15, 7, 36, 5, 10, 5, 4, 1, 1, 11, 10, 17, 8, 49, 6, 13, 7, 9, 2, 1, 1, 12, 11, 19, 9, 64, 7, 16, 9, 16, 3, 6, 1

Offset: 1

Gus Wiseman, Apr 28 2021

First differs from A343658 at A(4,2) = 5, A343658(4,2) = 6.

As a triangle, T(n,k) = number of divisors of k^(n-k).

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=1:  1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8
  n=3:  1   2   3   4   5   6   7   8
  n=4:  1   3   5   7   9  11  13  15
  n=5:  1   2   3   4   5   6   7   8
  n=6:  1   4   9  16  25  36  49  64
  n=7:  1   2   3   4   5   6   7   8
  n=8:  1   4   7  10  13  16  19  22
  n=9:  1   3   5   7   9  11  13  15
Triangle begins:
  1
  1  1
  1  2  1
  1  3  2  1
  1  4  3  3  1
  1  5  4  5  2  1
  1  6  5  7  3  4  1
  1  7  6  9  4  9  2  1
  1  8  7 11  5 16  3  4  1
  1  9  8 13  6 25  4  7  3  1
  1 10  9 15  7 36  5 10  5  4  1
  1 11 10 17  8 49  6 13  7  9  2  1
  1 12 11 19  9 64  7 16  9 16  3  6  1
  1 13 12 21 10 81  8 19 11 25  4 15  2  1
For example, row n = 8 counts the following divisors:
  1  64  243  256  125  36  7  1
     32  81   128  25   18  1
     16  27   64   5    12
     8   9    32   1    9
     4   3    16        6
     2   1    8         4
     1        4         3
              2         2
              1         1

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..9 of the array give A000005, A048691, A048785, A344327, A344328, A344329, A343526, A344335, A344336.
Row n = 6 of the array is A000290.
Diagonal n = k of the array is A062319.
Array antidiagonal sums (row sums of the triangle) are A343657.
Dominated by A343658.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998(n,k) = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
Cf. A000169, A000272, A002064, A002109, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996, A343652.

Table[DivisorSigma[0,k^(n-k)],{n,10},{k,n}]

A(n, k) = numdiv(n^k); \\ Seiichi Manyama, May 15 2021

A(n,k) = A000005(A009998(n,k)), where A009998(n,k) = n^k is the interpretation as an array.

A(n,k) = Sum_{d|n} k^omega(d). - Seiichi Manyama, May 15 2021

A239295 Number of words of length n over the alphabet {0,...,n-1} that avoid the pattern 123.

Original entry on oeis.org

1, 1, 4, 26, 210, 1897, 18368, 186636, 1965414, 21277685, 235493544, 2653779856, 30357956720, 351719984280, 4119552129280, 48708104589368, 580682799531822, 6973356315752445, 84286657672243880, 1024694111031383100, 12522664914160322460, 153762682439070435390

Offset: 0

Chad Brewbaker, Mar 14 2014

nonn

			a(0) = [].
a(1) = [0].
a(2) = [0,0], [0,1], [1,0], [1,1].
a(3) = [0,0,0], [0,0,1], [0,0,2], [0,1,0], [0,1,1], [0,2,0], [0,2,1], [0,2,2], [1,0,0], [1,0,1], [1,0,2], [1,1,0], [1,1,1], [1,1,2], [1,2,0], [1,2,1], [1,2,2], [2,0,0], [2,0,1], [2,0,2], [2,1,0], [2,1,1], [2,1,2], [2,2,0], [2,2,1], [2,2,2].

Alois P. Heinz, Table of n, a(n) for n = 0..800

Cf. A000108 (the permutation analog for 123-avoiding), A000312, A245667.

a:= proc(n) option remember; `if`(n<3, [1, 1, 4][n+1],
      ((7324*n^4-36350*n^3+58408*n^2-36126*n+8352) *a(n-1)
      -3*(n-3)*(2083*n^3-5374*n^2+2979*n+816) *a(n-2)
      -63*(n-3)*(n-4)*(3*n-7)*(3*n-8) *a(n-3)) /
      (4*n*(n-2)*(n+1)*(127*n-261)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 15 2014

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n-1, Table[Min[l[[j]], If[j == 1 || l[[j-1]] < i, i, l[[j]]]], {j, 1, Length[l]}]], {i, 1, l[[-1]]}]];
A[n_, k_] := A[n, k] = If[k == 0, If[n == 0, 1, 0], b[n, Array[n&, k]]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]];
a[n_] := Sum[T[n, k], {k, 0, 2}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 29 2017, after Alois P. Heinz (cf. A245667) *)

a(n) = Sum_{k=0..2} A245667(n,k).

a(n) ~ 3^(3*n-1/2) / (5^(3/2) * Pi * 2^(n-3) * n^2). - Vaclav Kotesovec, Sep 11 2014

a(8)-a(11) from Giovanni Resta, Mar 14 2014

a(12)-a(21) from Alois P. Heinz, Mar 15 2014

A247026 Number A(n,k) of endofunctions on [n] that are the k-th power of an endofunction; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 27, 1, 1, 1, 4, 12, 256, 1, 1, 1, 3, 19, 100, 3125, 1, 1, 1, 4, 12, 116, 1075, 46656, 1, 1, 1, 3, 21, 73, 985, 13356, 823543, 1, 1, 1, 4, 10, 148, 580, 11026, 197764, 16777216, 1, 1, 1, 3, 21, 44, 1281, 5721, 145621, 3403576, 387420489, 1

Offset: 0

Alois P. Heinz, Sep 09 2014

Number of endofunctions f on [n] such that an endofunction g on [n] exists with f=g^k.

			A(3,2) = 12: (1,1,1), (1,1,3), (1,2,1), (1,2,2), (1,2,3), (1,3,3), (2,2,2), (2,2,3), (2,3,1), (3,1,2), (3,2,3), (3,3,3).
A(3,6) = 10: (1,1,1), (1,1,3), (1,2,1), (1,2,2), (1,2,3), (1,3,3), (2,2,2), (2,2,3), (3,2,3), (3,3,3).
A(4,4) = 73: (1,1,1,1), (1,1,1,4), (1,1,3,1), (1,1,3,3), ..., (4,4,1,3), (4,4,2,3), (4,4,3,4), (4,4,4,4).
Square array A(n,k) begins:
  1,      1,      1,      1,     1,      1,     1,      1, ...
  1,      1,      1,      1,     1,      1,     1,      1, ...
  1,      4,      3,      4,     3,      4,     3,      4, ...
  1,     27,     12,     19,    12,     21,    10,     21, ...
  1,    256,    100,    116,    73,    148,    44,    148, ...
  1,   3125,   1075,    985,   580,   1281,   295,   1305, ...
  1,  46656,  13356,  11026,  5721,  12942,  3136,  13806, ...
  1, 823543, 197764, 145621, 69244, 150955, 42784, 169681, ...

Columns k=0-10 give: A000012, A000312, A102687, A163859, A163860, A163861, A247053, A247054, A247055, A247056, A247057.
Rows n=0+1, 2-7 give: A000012, A103947, A103948, A103949, A102709, A103950, A247058.
Main diagonal gives A247059.
Cf. A247005 (the same for permutations).

(* This program is not suitable to compute a large number of terms. *)
nmax = 8;
f[a_][b_] /; Length[a]==Length[b] := Table[b[[a[[i]]]], {i, 1, Length[a]}];
A[n_, k_] := Nest[f[#], Range[n], k]& /@ Tuples[Range[n], {n}] // Union // Length;
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *)

A264902 Number T(n,k) of defective parking functions of length n and defect k; triangle T(n,k), n>=0, 0<=k<=max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 3, 1, 16, 10, 1, 125, 107, 23, 1, 1296, 1346, 436, 46, 1, 16807, 19917, 8402, 1442, 87, 1, 262144, 341986, 173860, 41070, 4320, 162, 1, 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1, 100000000, 148717762, 96920092, 34268902, 6768184, 710314, 34660, 574, 1

Offset: 0

Alois P. Heinz, Nov 28 2015

			T(2,0) = 3: [1,1], [1,2], [2,1].
T(2,1) = 1: [2,2].
T(3,1) = 10: [1,3,3], [2,2,2], [2,2,3], [2,3,2], [2,3,3], [3,1,3], [3,2,2], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 1: [3,3,3].
Triangle T(n,k) begins:
0 :       1;
1 :       1;
2 :       3,       1;
3 :      16,      10,       1;
4 :     125,     107,      23,       1;
5 :    1296,    1346,     436,      46,      1;
6 :   16807,   19917,    8402,    1442,     87,     1;
7 :  262144,  341986,  173860,   41070,   4320,   162,   1;
8 : 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1;
    ...

Alois P. Heinz, Rows n = 0..141, flattened
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008

Columns k=0-10 give: A000272(n+1), A140647, A291128, A291129, A291130, A291131, A291132, A291133, A291134, A291135, A291136.
Row sums give A000312.
T(2n,n) gives A264903.
Cf. A036276, A101334, A274279.

S:= (n, k)-> `if`(k=0, n^n, add(binomial(n, i)*k*
            (k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k)):
T:= (n, k)-> S(n, k)-S(n, k+1):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..10);

S[n_, k_] := If[k==0, n^n, Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}]]; T[n_, k_] := S[n, k]-S[n, k+1]; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, Max[0, n-1]}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

T(n,k) = S(n,k) - S(n,k+1) with S(n,0) = n^n, S(n,k) = Sum_{i=0..n-k} C(n,i) * k*(k+i)^(i-1) * (n-k-i)^(n-i) for k>0.
Sum_{k>0} k * T(n,k) = A036276(n-1) for n>0.
Sum_{k>0} T(n,k) = A101334(n).
Sum_{k>=0} (-1)^k * T(n,k) = A274279(n) for n>=1.

A275549 Number of classes of endofunctions of [n] under reversal.

Original entry on oeis.org

1, 1, 3, 18, 136, 1625, 23436, 412972, 8390656, 193739769, 5000050000, 142656721086, 4458051717120, 151437584670385, 5556003465485760, 218946946471875000, 9223372039002259456, 413620131002462320337, 19673204037747448432896, 989209827833222327690890

Offset: 0

Olivier Gérard, Aug 01 2016

f and g are in the same class if function g(i) = f(n+1-i) for all i.
Decomposition by class size
.
n    1      2
---------------
1    1      0
2    2      1
3    9      9
4    16     120
5    125    1500
6    216    23220
7    2401   410571
.
Demonstration for the formula: the classes are either of size 1 or 2.
The classes of size 1 is for functions invariant by reversal. They are specified by half their values, including one more if n is odd. Their number is n^(ceiling(n/2)).
So the number of classes under this symmetry is half (the number of functions + the number of classes of size 1).
a(n) is the number of unoriented length n strings with a maximum of n colors. - Andrew Howroyd, Sep 13 2019

Andrew Howroyd, Table of n, a(n) for n = 0..200

Main diagonal of A277504.
Cf. A000312 All endofunctions
Cf. A000169 Classes under translation mod n
Cf. A001700 Classes under sort
Cf. A056665 Classes under rotation
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
Cf. A081721 Classes under rotation and reversal
Cf. A275550 Classes under reversal and complement
Cf. A275551 Classes under translation and reversal
Cf. A275552 Classes under translation and complement
Cf. A275553 Classes under translation, complement and reversal
Cf. A275554 Classes under translation, rotation and complement
Cf. A275555 Classes under translation, rotation and reversal
Cf. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal
Cf. A078707 Endofunctions symmetric around their middle (stable by reversal).

a(n) = {(n^n + n^((n+1)\2))/2} \\ Andrew Howroyd, Sep 13 2019

a(n) = (n^n+n^ceiling(n/2))/2.

A031973 a(n) = Sum_{k=0..n} n^k.

Original entry on oeis.org

1, 2, 7, 40, 341, 3906, 55987, 960800, 19173961, 435848050, 11111111111, 313842837672, 9726655034461, 328114698808274, 11966776581370171, 469172025408063616, 19676527011956855057, 878942778254232811938, 41660902667961039785743, 2088331858752553232964200

Offset: 0

N. J. A. Sloane

These are the generalized repunits of length n+1 in base n for all n >= 1: a(n) expressed in base n is 111...111 (n+1 1's): a(1) = 1^0 + 1^1 = 2 = A000042(2), a(2) = 2^0 + 2^1 + 2^2 = 7 = A000225(3), a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40 = A003462(4), etc., a(10) = 10^0 + 10^1 + 10^2 + ... + 10^9 + 10^10 = 11111111111 = A002275(11), etc. - Rick L. Shepherd, Aug 26 2004
a(n)=the total number of ordered selections of up to n objects from n types with repetitions allowed.  Thus for 2 objects a,b there are 7 possible selections: aa,bb,ab,ba,a,b, and the null set. - J. M. Bergot, Mar 26 2014
a(n)=the total number of ordered arrangements of 0,1,2..n objects, with repetitions allowed, selected from n types of objects. - J. M. Bergot, Apr 11 2014

			a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40.

Nathaniel Johnston, Table of n, a(n) for n = 0..100

Cf. A000042 (unary representations), A000225 (2^n-1: binary repunits shown in decimal), A003462 ((3^n-1)/2: ternary repunits shown in decimal), A002275 ((10^n-1)/9: decimal repunits).

Cf. A104878.

[&+[n^k: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

a:= proc(n) local c, i; c:=1; for i to n do c:= c*n+1 od; c end:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 15 2013

Join[{1},Table[Total[n^Range[0,n]],{n,20}]] (* Harvey P. Dale, Nov 13 2011 *)

a(n)=(n^(n+1)-1)/(n-1) \\ Charles R Greathouse IV, Mar 26 2014

[lucas_number1(n,n,n-1) for n in range(1, 19)] # Zerinvary Lajos, May 16 2009

a(n) = (n^(n+1)-1)/(n-1) = (A007778(n)-1)/(n-1) = A023037(n)+A000312(n) = A031972(n)+1. - Henry Bottomley, Apr 04 2003
a(n) = A125118(n,n-2) for n>2. - Reinhard Zumkeller, Nov 21 2006
a(n) = [x^n] 1/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017
a(n) = A104878(2n,n). - Alois P. Heinz, May 04 2021

A062970 a(n) = 1 + Sum_{j=1..n} j^j.

Original entry on oeis.org

1, 2, 6, 33, 289, 3414, 50070, 873613, 17650829, 405071318, 10405071318, 295716741929, 9211817190185, 312086923782438, 11424093749340454, 449317984130199829, 18896062057839751445, 846136323944176515622, 40192544399240714091046, 2018612200059554303215025

Offset: 0

Henry Bottomley, Jul 23 2001

nonn

The usual convention in the OEIS is that 0^0 = 1. This sequence could therefore be defined as Sum_{j=0..n} j^j. See also A001923.

			a(4) = 1 + 1^1 + 2^2 + 3^3 + 4^4 = 1 + 1 + 4 + 27 + 256 = 289.

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A000312, A001923.

Table[Sum[Sum[Binomial[n, k] StirlingS2[n, k] k!, {k, 0, n}], {n, 0, m}], {m, 0, 20}] (* Geoffrey Critzer, Mar 18 2009 *)
Join[{1},Accumulate[Table[n^n,{n,20}]]+1] (* Harvey P. Dale, Aug 31 2016 *)

{ a=0; for (n=0, 100, write("b062970.txt", n, " ", a+=n^n) ) } \\ Harry J. Smith, Aug 14 2009

from itertools import count, accumulate, islice
def A062970_gen(): # generator of terms
    yield from accumulate((k**k for k in count(1)),initial=1)
A062970_list = list(islice(A062970_gen(),20)) # Chai Wah Wu, Jun 17 2022

a(n) = a(n-1) + A000312(n) = A001923(n) + 1.

a(n) ~ n^n. - Vaclav Kotesovec, Nov 27 2017

A079901 Triangle of powers, T(n,k) = n^k, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 27, 1, 4, 16, 64, 256, 1, 5, 25, 125, 625, 3125, 1, 6, 36, 216, 1296, 7776, 46656, 1, 7, 49, 343, 2401, 16807, 117649, 823543, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721

Offset: 0

Reinhard Zumkeller, Feb 21 2003

Matrix inverse equals the triangle R where R(n,k) = A107045(n,k)/A107046(n,k) are coefficients with exponential-like properties. - Paul D. Hanna, May 22 2005

			Triangle begins:
  1;
  1,1;
  1,2,4;
  1,3,9,27;
  1,4,16,64,256;
  1,5,25,125,625,3125;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A107045/A107046 (matrix inverse).

Cf. A002262, A000290, A000578, A000583, A000272, A000169, A000312.

a079901 n k = a079901_tabl !! n !! k
a079901_row n = a079901_tabl !! n
a079901_tabl = zipWith (map . (^)) [0..] a002262_tabl
-- Reinhard Zumkeller, Mar 31 2015

Join[{1},Flatten[Table[n^k,{n,9},{k,0,n}]]] (* Harvey P. Dale, Feb 08 2013 *)

row(n) = vector(n+1, k, n^(k-1)); \\ Amiram Eldar, May 09 2025

T(n,k) = if k=0 then 1 else T(n,k-1)*n.
T(n,0) = 1; T(n,1) = n for n>0; T(n,2) = A000290(n) for n > 1; T(n,3) = A000578(n) for n > 2; T(n,4) = A000583(n) for n>3.
T(n,n-2) = A000272(n) for n>2; T(n,n-1) = A000169(n) for n>1; T(n,n) = A000312(n).

A168658 a(n) = ceiling(n^n/2).

Original entry on oeis.org

1, 1, 2, 14, 128, 1563, 23328, 411772, 8388608, 193710245, 5000000000, 142655835306, 4458050224128, 151437553296127, 5556003412779008, 218946945190429688, 9223372036854775808, 413620130943168382089

Offset: 0

Zerinvary Lajos, Dec 02 2009

Number of functions of [n] to [n] (endofunctions of degree n) up to complement to n+1.

There is only one function, and only when n=2k-1 is odd, fixed by n+1-complement, the constant function with value k.

			Ceiling(6^6/2) = 23328.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000312 (all endofunctions of degree n)

Cf. A057065 (floor of n^n / 2).

[Ceiling(n^n/2): n in [0..20]]; // Vincenzo Librandi, Aug 29 2011

Join[{1}, Table[Ceiling[n^n/2], {n, 1, 25}]] (* G. C. Greubel, Jul 28 2016 *)

a(n) = ceil(n^n/2); \\ Michel Marcus, Feb 18 2016

Sage
```
[ceil(n^n/2) for n in range(0,21)]#
```

a(n) = ceiling(A000312(n)/2).

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