A239001 -id:A239001 - cp's OEIS frontend

A239512 Irregular triangular array read by rows: row n gives a list of the partitions of the Lucas numbers.

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 4, 1, 1, 3, 3, 3, 2, 1, 3, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 4, 3, 4, 2, 1, 4, 1, 1, 1, 3, 3, 1

Offset: 1

Clark Kimberling, Mar 25 2014

The number of partitions represented in row n is A067592(n). The parts of each partition are arranged in nonincreasing order, and the partitions are arranged in Mathematica order (reverse-lexicographic). The parts are the terms of the Lucas sequence, A000032(n), n >= 1.

			The first 7 rows:
1
1 1
3 1 1 1
4 3 1 1 1 1 1
4 1 3 1 1 1 1 1 1 1
4 1 1 3 3 3 1 1 1 1 1 1 1 1 1
7 4 3 4 1 1 1 3 3 1 3 1 1 1 1 1 1 1 1 1 1 1
The first 7 rows represent these partitions:
1
11
3, 111
4, 31, 1111
41, 311, 11111
411, 33, 3111, 111111
7, 43, 431, 41111, 3311, 311111, 1111111

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A239001, A000032, A067592.

LucasQ[n_] := IntegerQ[Sqrt[5 n^2 + 20]] || IntegerQ[Sqrt[5 n^2 - 20]];
Attributes[LucasQ] = {Listable}; TableForm[t = Map[Select[IntegerPartitions[#], And @@ LucasQ[#] &] &, Range[0, 12]]]  (* A239512, partitions *)
Flatten[t] (* A067592 *)
(* Peter J. C. Moses, Mar 24 2014 *)

A240224 Irregular triangular array read by rows: row n gives a list of the partitions of n into distinct Fibonacci numbers. The order of the partitions is like in Abramowitz-Stegun.

Original entry on oeis.org

1, 2, 3, 2, 1, 3, 1, 5, 3, 2, 5, 1, 3, 2, 1, 5, 2, 8, 5, 3, 5, 2, 1, 8, 1, 5, 3, 1, 8, 2, 5, 3, 2, 8, 3, 8, 2, 1, 5, 3, 2, 1, 8, 3, 1, 13, 8, 5, 8, 3, 2, 13, 1, 8, 5, 1, 8, 3, 2, 1, 13, 2, 8, 5, 2, 13, 3, 13, 2, 1, 8, 5, 3, 8, 5, 2, 1, 13, 3, 1, 8, 5, 3, 1, 13, 5, 13, 3, 2, 8, 5, 3, 2, 13, 5, 1, 13, 3, 2, 1, 8, 5, 3, 2, 1, 13, 5, 2

Offset: 1

Wolfdieter Lang, Apr 07 2014

The row length sequence is A240225. The number of partitions in row n is A000119(n).

The order of the partitions is like in Abramowitz-Stegun (rising number of parts, within like part numbers lexicographic) but here the order of the parts has been reversed, that is they are ordered decreasingly.

			The array with separated partitions begins:
n\k       1         2          3         4            5 ...
1:        1
2:        2
3:        3       2,1
4:      3,1
5:        5       3,2
6:      5,1     3,2,1
7:      5,2
8:        8       5,3      5,2,1
9:      8,1     5,3,1
10:     8,2     5,3,2
11:     8,3     8,2,1    5,3,2,1
12:   8,3,1
13:      13       8,5      8,3,2
14:    13,1     8,5,1    8,3,2,1
15:    13,2     8,5,2
16:    13,3    13,2,1      8,5,3   8,5,2,1
17:  13,3,1   8,5,3,1
18:    13,5    13,3,2    8,5,3,2
19:  13,5,1  13,3,2,1  8,5,3,2,1
20:  13,5,2
21:      21      13,8     13,5,3  13,5,2,1
22:    21,1    13,8,1   13,5,3,1
23:    21,2    13,8,2   13,5,3,2
24:    21,3    21,2,1     13,8,3  13,8,2,1   13,5,3,2,1
25:  21,3,1  13,8,3,1
...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000119, A239001, A240225.

cp's OEIS Frontend

A239512 Irregular triangular array read by rows: row n gives a list of the partitions of the Lucas numbers.

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