A176210 -id:A176210 - cp's OEIS frontend

A176211 Numbers of the form Product_{m_i >= 3} A000211(m_i), possibly repeated, in natural order.

6, 9, 13, 20, 31, 36, 49, 54, 78, 78, 81, 117, 120, 125, 169, 180, 186, 201, 216, 260, 279, 294, 324, 324, 400, 403, 441, 468, 468, 486, 523, 620, 637, 702, 702, 720, 729, 750, 845, 961, 980, 1014, 1014, 1053, 1080, 1116, 1125, 1206, 1296, 1366, 1519, 1521, 1560, 1560, 1620, 1625, 1674, 1764, 1809, 1944, 1944, 2197, 2209

Offset: 1

Vladimir Shevelev, Apr 12 2010

nonn

Values represented by more than one set of indices are listed once per set; otherwise A176212 results.
Each term is a permanent of a quadratic symmetric (0,1)  matrix with 1's on the main diagonal and exactly three 1's in each row and column.
For fixed Sum m_i=n with m_i >= 3, Product A000211(m_i) >= 6(4/3)^(n-3) and max(Product A000211(m_i)) = 6^((n-h)/3)*floor((3/2)^h), where h is the remainder of n (mod 3).

V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317.

Cf. A176210, A036036, A036037, A080577, A138136, A000211.

f(n) = fibonacci(n+1) + fibonacci(n-1) + 2; \\ A000211
lista(nn) = {my(v = vector(nn, k, f(k+2))); my(vmax = vecmax(v)); my(w =  vector(nn, k, [0, logint(vmax, v[k])])); my(list=List()); forvec(x = w, if (vecmax(x), my(y = prod(k=1, #v, v[k]^x[k])); if (y <= vmax, listput(list, y)););); Vec(vecsort(list));}
lista(14) \\ Michel Marcus, Jan 06 2021

A176212 Terms of A176211, duplicates removed.

Original entry on oeis.org

6, 9, 13, 20, 31, 36, 49, 54, 78, 81, 117, 120, 125, 169, 180, 186, 201, 216, 260, 279, 294, 324, 400, 403, 441, 468, 486, 523, 620, 637, 702, 720, 729, 750, 845, 961, 980, 1014, 1053, 1080, 1116, 1125, 1206, 1296, 1366, 1519, 1521, 1560, 1620, 1625, 1674, 1764, 1809, 1944, 2197

Offset: 1

Vladimir Shevelev, Apr 12 2010

nonn

The terms are permanents of a set of certain symmetric (0,1)-matrices as detailed in A176211. Thus the sequence solves a symmetric version of Gristein problem: to find all the values of permanent of all square (0,1) matrices, which contain exactly three 1's in each row and column (see the list of unsolved problems in chapter 8 of Minc's book).

H. Minc, Permanents, Addison-Wesley, 1978.

V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317

Cf. A176211, A176210, A000211.

f(n) = fibonacci(n+1) + fibonacci(n-1) + 2; \\ A000211
lista(nn) = {my(v = vector(nn, k, f(k+2))); my(vmax = vecmax(v)); my(w =  vector(nn, k, [0, logint(vmax, v[k])])); my(list=List()); forvec(x = w, if (vecmax(x), my(y = prod(k=1, #v, v[k]^x[k])); if (y <= vmax, listput(list, y)););); Vec(vecsort(list,,8));}
lista(14) \\ Michel Marcus, Jan 06 2021

A177740 Triangle read by rows in which row n lists the number of parts in each of those partitions of n that have every part >=3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 4, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 4, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 5

Offset: 3

Jason Kimberley, May 13 2010

The ordering of the partitions is described in A176210.

			3: 3 (1)
4: 4 (1)
5: 5 (1)
6: 6 or 3,3 (1, 2)
7: 7 or 3,4 (1, 2)
8: 8, 3,5 or 4,4 (1, 2, 2)
so sequence begins 1, 1, 1, 1,2, 1,2, ...

J. S. Kimberley, Rows 3..39 of A177740 triangle, flattened.

The length of the rows is given by A008483.

Cf. A128628 for unrestricted partitions.

[ #part: part in RestrictedPartitions(n,{3..n}), n in [0..16]];

Edited (including b-file) by N. J. A. Sloane, Oct 05 2010

A177741 Triangle in which row n lists the numbers of strong vertex magic total labelings of each 2-regular simple graph on 2n+1 vertices.

Original entry on oeis.org

1, 1, 9, 0, 31, 8, 4, 2, 308, 81, 100, 70, 0, 7, 3809, 578, 474, 495, 454, 103, 181, 103, 97, 0, 63995, 11703, 11655, 9472, 9252, 1151, 8567, 2297, 1758, 1389, 1117, 2023, 104, 328, 210, 128, 11, 1152784, 201685, 193899, 159485, 144516, 19625, 137561, 38453

Offset: 1

Jason Kimberley, May 17 2010

The 2-regular simple graphs are the disjoint unions of simple cycles (the smallest simple cycle being a triangle).
A simple counting argument shows that no 2-regular graph of even order possesses a strong VMTL.
The ordering of the graphs in row n is the ordering of the corresponding partitions listed in row 2n+1 of A176210.

J. S. Kimberley, Rows 1..9 of A177741 triangle, flattened.
I. D. Gray and J. A. MacDougall, Vertex-magic labelings of regular graphs. II, Discrete Mathematics, 309 (2009), 5986-5999.

The length of row n is A008483(2n+1).
The row sums are A177742.
The first column is A177743.

A177739 In those partitions of n with every part >=3, the total number of parts (counted with multiplicity).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 5, 8, 10, 13, 22, 25, 34, 49, 62, 77, 108, 132, 172, 221, 276, 345, 448, 544, 680, 851, 1050, 1280, 1596, 1931, 2366, 2884, 3496, 4220, 5135, 6144, 7403, 8890, 10644, 12679, 15177, 18007, 21419, 25399, 30066, 35488, 41971, 49344, 58088

Offset: 0

Jason Kimberley, May 13 2010

Also the number of components (counted with multiplicity) of the 2-regular simple graphs of order n.

The number of such partitions is given by A008483.
Lengths of the rows of triangle A176210.
Row sums of triangle A177740.
Cf. A006128, A138135.

[ #&cat RestrictedPartitions(n,{3..n}):n in [0..50]];

Table[Length[Flatten[Select[IntegerPartitions[n],Min[#]>2&]]],{n,0,50}] (* Harvey P. Dale, May 12 2020 *)

cp's OEIS Frontend

A176211 Numbers of the form Product_{m_i >= 3} A000211(m_i), possibly repeated, in natural order.

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A176212 Terms of A176211, duplicates removed.

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A177740 Triangle read by rows in which row n lists the number of parts in each of those partitions of n that have every part >=3.

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A177741 Triangle in which row n lists the numbers of strong vertex magic total labelings of each 2-regular simple graph on 2n+1 vertices.

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Author

Keywords

Comments

Links

Crossrefs

A177739 In those partitions of n with every part >=3, the total number of parts (counted with multiplicity).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica