A241842 -id:A241842 - cp's OEIS frontend

A241740 Number of partitions p of n such that (number of numbers in p of form 3k+2) < (number of numbers in p of form 3k).

0, 0, 0, 1, 1, 1, 3, 4, 4, 7, 10, 12, 17, 24, 30, 40, 53, 70, 90, 118, 152, 194, 244, 316, 396, 497, 626, 784, 960, 1202, 1483, 1816, 2230, 2738, 3312, 4042, 4908, 5922, 7141, 8627, 10327, 12388, 14832, 17703, 21075, 25120, 29795, 35321, 41822, 49439, 58286

Offset: 0

Clark Kimberling, Apr 28 2014

Each number in p is counted once, regardless of its multiplicity.

			a(8) counts these 4 partitions:  611, 431, 3311, 311111.

Cf. A241737, A241741, A241741, A241743.

z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[2, p] < s[0, p]], {n, 0, z}]  (* A241740 *)
Table[Count[f[n], p_ /; s[2, p] == s[0, p]], {n, 0, z}] (* A241741 *)
Table[Count[f[n], p_ /; s[2, p] > s[0, p]], {n, 0, z}]  (* A241742 *)

a(n) + A241741(n) + A241842(n) = A000041(n) for n >= 0.

A241741 Number of partitions p of n such that (number of numbers in p of form 3k+2) = (number of numbers in p of form 3k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 5, 9, 11, 14, 22, 29, 36, 51, 66, 83, 107, 139, 170, 216, 273, 340, 415, 520, 635, 778, 952, 1177, 1414, 1724, 2094, 2527, 3038, 3691, 4411, 5286, 6345, 7586, 9008, 10778, 12796, 15163, 17979, 21288, 25059, 29608, 34861, 40927, 48035

Offset: 0

Clark Kimberling, Apr 28 2014

Each number in p is counted once, regardless of its multiplicity.

			a(8) counts these 9 partitions:  71, 62, 53, 44, 41111, 332, 3221, 32111, 11111111.

Cf. A241737, A241740, A241742, A241743.

z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[2, p] < s[0, p]], {n, 0, z}]  (* A241740 *)
Table[Count[f[n], p_ /; s[2, p] == s[0, p]], {n, 0, z}] (* A241741 *)
Table[Count[f[n], p_ /; s[2, p] > s[0, p]], {n, 0, z}]  (* A241742 *)

a(n) + A241740(n) + A241842(n) = A000041(n) for n >= 0.

A243785 Number of unlabeled simple connected graphs with n nodes that are chordal but not integral.

Original entry on oeis.org

0, 0, 1, 4, 12, 56, 267, 1605, 11909, 109525

Offset: 1

Travis Hoppe and Anna Petrone, Jun 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs.
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Chordal Graph.
Eric Weisstein's World of Mathematics, Integral Graph.

Cf. A048192 (connected chordal graphs), A241842 (non-integral graphs).

Cf. A243786.

a(n) = A048192(n) - A243786(n).

A243274 Number of graphs with n nodes that are Hamiltonian and non-integral.

Original entry on oeis.org

1, 0, 1, 2, 1, 5, 4, 11, 12, 50

Offset: 1

Travis Hoppe and Anna Petrone, Jun 02 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014

Cf. A003216 (Hamiltonian graphs), A064731 (integral graphs), A241842 (non-integral graphs).

A243323 Number of simple connected graphs with n nodes that are bipartite and not integral.

Original entry on oeis.org

0, 0, 1, 2, 4, 14, 43, 179, 730, 4019

Offset: 1

Travis Hoppe and Anna Petrone, Jun 03 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014

Cf. A003216 (bipartite graphs), A241842 (non-integral graphs).

A243324 Number of simple connected graphs with n nodes that are Eulerian and not integral.

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 33, 180, 1773, 31006

Offset: 1

Travis Hoppe and Anna Petrone, Jun 03 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014

Cf. A003049 (Eulerian graphs), A241842 (non-integral graphs).

A243325 Number of simple connected graphs with n nodes that are planar and not integral.

Original entry on oeis.org

0, 0, 1, 4, 18, 95, 642, 5962, 71876, 1052786

Offset: 1

Travis Hoppe and Anna Petrone, Jun 03 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.

Cf. A003094 (planar graphs), A241842 (non-integral graphs).

A243326 Number of simple connected graphs with n nodes that are triangle-free and not integral.

Original entry on oeis.org

0, 0, 1, 2, 5, 16, 58, 264, 1380, 9818

Offset: 1

Travis Hoppe and Anna Petrone, Jun 03 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.

Cf. A241842 (non-integral graphs), A024607 (triangle-free graphs).

A243327 Number of simple connected graphs with n nodes that are K_4 free and not integral.

Original entry on oeis.org

0, 0, 1, 4, 15, 77, 531, 5597, 95900, 2784034

Offset: 1

Travis Hoppe and Anna Petrone, Jun 03 2014

K_4 is the complete graph on four vertices.

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.

Cf. A241842 (non-integral graphs), A079574 (K_4-free graphs).

A243338 Number of simple connected graphs with n nodes that are trees and not integral.

Original entry on oeis.org

0, 0, 1, 2, 2, 5, 10, 23, 47, 105

Offset: 1

Travis Hoppe and Anna Petrone, Jun 03 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014

Cf. A241842 (non-integral graphs), A000055 (tree graphs).

cp's OEIS Frontend

A241740 Number of partitions p of n such that (number of numbers in p of form 3k+2) < (number of numbers in p of form 3k).

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A241741 Number of partitions p of n such that (number of numbers in p of form 3k+2) = (number of numbers in p of form 3k).

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A243785 Number of unlabeled simple connected graphs with n nodes that are chordal but not integral.

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A243274 Number of graphs with n nodes that are Hamiltonian and non-integral.

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A243323 Number of simple connected graphs with n nodes that are bipartite and not integral.

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A243324 Number of simple connected graphs with n nodes that are Eulerian and not integral.

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A243325 Number of simple connected graphs with n nodes that are planar and not integral.

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A243326 Number of simple connected graphs with n nodes that are triangle-free and not integral.

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A243327 Number of simple connected graphs with n nodes that are K_4 free and not integral.

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A243338 Number of simple connected graphs with n nodes that are trees and not integral.

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