A323872
Number of n X n aperiodic binary toroidal necklaces.
Original entry on oeis.org
1, 2, 2, 54, 4050, 1342170, 1908852102, 11488774559598, 288230375950387200, 29850020237398244599296, 12676506002282260237970435130, 21970710674130840874443091905460038, 154866286100907105149455216472736043777350, 4427744605404865645682169434028029029963535277450
Offset: 0
Inequivalent representatives of the a(2) = 2 aperiodic necklaces:
[0 0] [0 1]
[0 1] [1 1]
Inequivalent representatives of the a(3) = 54 aperiodic necklaces:
000 000 000 000 000 000 000 000 000
000 000 001 001 001 001 001 001 001
001 011 001 010 011 100 101 110 111
.
000 000 000 000 000 000 000 000 000
011 011 011 011 011 011 011 111 111
001 010 011 100 101 110 111 001 011
.
001 001 001 001 001 001 001 001 001
001 001 001 001 001 001 010 010 010
010 011 100 101 110 111 011 101 110
.
001 001 001 001 001 001 001 001 001
010 011 011 011 011 011 100 100 100
111 010 011 101 110 111 011 110 111
.
001 001 001 001 001 001 001 001 001
101 101 101 101 110 110 110 110 111
011 101 110 111 011 101 110 111 011
.
001 001 001 011 011 011 011 011 011
111 111 111 011 011 011 101 110 111
101 110 111 101 110 111 111 111 111
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apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],And[apermatQ[#],neckmatQ[#]]&]],{n,4}]
A323868
Number of matrices of size n whose entries cover an initial interval of positive integers.
Original entry on oeis.org
1, 6, 26, 225, 1082, 18732, 94586, 2183340, 21261783, 408990252, 3245265146, 168549405570, 1053716696762, 42565371881772, 921132763911412, 26578273409906775, 260741534058271802, 20313207979541071938, 185603174638656822266, 16066126777466305218690
Offset: 1
The 42 matrices of size 4 whose entries cover {1,2}:
1222 2111 1122 2211 1212 2121 1221 2112 1112 2221 1121 2212 1211 2122
.
12 21 11 22 12 21 12 21 11 22 11 22 12 21
22 11 22 11 12 21 21 12 12 21 21 12 11 22
.
1 2 1 2 1 2 1 2 1 2 1 2 1 2
2 1 1 2 2 1 2 1 1 2 1 2 2 1
2 1 2 1 1 2 2 1 1 2 2 1 1 2
2 1 2 1 2 1 1 2 2 1 1 2 1 2
The 18 matrices of size 4 whose entries cover {1,2} with multiplicities {2,2}:
[1 1 2 2] [2 2 1 1] [1 2 1 2] [2 1 2 1] [1 2 2 1] [2 1 1 2]
.
[1 1] [2 2] [1 2] [2 1] [1 2] [2 1]
[2 2] [1 1] [1 2] [2 1] [2 1] [1 2]
.
[1] [2] [1] [2] [1] [2]
[1] [2] [2] [1] [2] [1]
[2] [1] [1] [2] [2] [1]
[2] [1] [2] [1] [1] [2]
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*binomial(n, j), j=1..n))
end:
a:= n-> b(n)*numtheory[tau](n):
seq(a(n), n=1..20); # Alois P. Heinz, Feb 04 2019
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
Table[Length[nrmmats[n]],{n,6}]
Table[DivisorSigma[0, n]*Sum[k! StirlingS2[n, k], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Feb 05 2019 *)
-
a(n) = numdiv(n)*sum(k=0, n, stirling(n, k, 2)*k!); \\ Michel Marcus, Feb 05 2019
A324914
a(n) = Sum_{k=1..n} 2^k * tau(k), where tau(k) = A000005(k).
Original entry on oeis.org
2, 10, 26, 74, 138, 394, 650, 1674, 3210, 7306, 11402, 35978, 52362, 117898, 248970, 576650, 838794, 2411658, 3460234, 9751690, 18140298, 34917514, 51694730, 185912458, 286575754, 555011210, 1091882122, 2702494858, 3776236682, 12366171274
Offset: 1
-
ListTools:-PartialSums([seq(2^k*numtheory:-tau(k),k=1..100)]); # Robert Israel, Jun 27 2019
-
Accumulate[Table[2^k*DivisorSigma[0, k], {k, 1, 30}]]
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