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A239001 Irregular triangular array read by rows: row n gives a list of the partitions of n into Fibonacci numbers. 3
1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 5, 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, 5, 2, 5, 1, 1, 3, 3, 1, 3, 2, 2, 3, 2, 1, 1, 3, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The number of partitions represented in row n is A003107(n).

The parts of a partition are nonincreasing and the order of the partitions is anti-lexicographic. As parts one uses A000045(n), n >= 2. - Wolfdieter Lang, Mar 17 2014

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

1

2 1 1

3 2 1 1 1 1

3 1 2 2 2 1 1 1 1 1 1

5 3 2 3 1 1 2 2 1 2 1 1 1 1 1 1 1 1

Row 5 represents these six partitions: 5, 32, 311, 221, 2111, 11111.

From Wolfdieter Lang, Mar 17 2014: (Start)

The array with separated partitions begins:

n\k   1      2      3        4        5          6          7            8             9             10 ...

1:    1

2:    2    1,1

3:    3    2,1  1,1,1

4:  3,1    2,2  2,1,1  1,1,1,1

5:    5    3,2  3,1,1    2,2,1  2,1,1,1  1,1,1,1,1

6:  5,1    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:  5,2  5,1,1  3,3,1    3,2,2  3,2,1,1  3,1,1,1,1    2,2,2,1    2,2,1,1,1   2,1,1,1,1,1  1,1,1,1,1,1,1

...

Row n=8: 8  5,3  5,2,1  5,1,1,1  3,3,2  3,3,1,1  3,2,2,1  3,2,1,1,1  3,1,1,1,1,1   2,2,2,2   2,2,2,1,1

  2,2,1,1,1,1  2,1,1,1,1,1,1  1,1,1,1,1,1,1,1;

Row n=9  8,1  5,3,1  5,2,2   5,2,1,1   5,1,1,1,1  3,3,3   3,3,2,1   3,3,1,1,1  3,2,2,2  3,2,2,1,1

3,2,1,1,1,1   3,1,1,1,1,1,1  2,2,2,2,1  2,2,2,1,1,1  2,2,1,1,1,1,1   2,1,1,1,1,1,1,1   1,1,1,1,1,1,1,1,1;

Row n=10: 8,2  8,1,1   5,5   5,3,2  5,3,1,1  5,2,2,1  5,2,1,1,1  5,1,1,1,1,1   3,3,3,1  3,3,2,2  3,3,2,1,1

  3,3,1,1,1,1   3,2,2,2,1  3,2,2,1,1,1   3,2,1,1,1,1,1   3,1,1,1,1,1,1,1   2,2,2,2,2   2,2,2,2,1,1

  2,2,2,1,1,1,1  2,2,1,1,1,1,1,1  2,1,1,1,1,1,1,1,1  1,1,1,1,1,1,1,1,1,1.

-----------------------------------------------------------------------------------------------------------

(End)

MATHEMATICA

f = Table[Fibonacci[n], {n, 2, 60}]; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; s[n_, k_] := If[Union[f, DeleteDuplicates[p[n, k]]] == f, p[n, k], 0]; t[n_] := Table[s[n, k], {k, 1, PartitionsP[n]}]; TableForm[Table[DeleteCases[t[n], 0], {n, 1, 12}]] (* shows partitions *)

y = Flatten[Table[DeleteCases[t[n], 0], {n, 1, 12}]] (* A239001 *)

(* also *)

FibonacciQ[n_] := IntegerQ[Sqrt[5 n^2 + 4]] || IntegerQ[Sqrt[5 n^2 - 4]]; Attributes[FibonacciQ] = {Listable}; TableForm[t = Map[Select[IntegerPartitions[#], And @@ FibonacciQ[#] &] &, Range[0, 12]]]

Flatten[t] (* Peter J. C. Moses, Mar 24 2014 *)

CROSSREFS

Cf. A003107, A000045, A239512.

Sequence in context: A190688 A145975 A211028 * A277648 A026792 A139100

Adjacent sequences:  A238998 A238999 A239000 * A239002 A239003 A239004

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Mar 08 2014

STATUS

approved

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Last modified June 20 09:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)