A335712 The sum of the sizes of the minimal fixed points over all compositions of n.

1, 1, 2, 6, 12, 27, 54, 115, 237, 486, 997, 2030, 4122, 8350, 16881, 34054, 68609, 138052, 277500, 557328, 1118546, 2243589, 4498004, 9014053, 18058159, 36166338, 72415886, 144970116, 290170091, 580721926, 1162077483, 2325206168, 4652155420, 9307199819

Offset

1,3

References

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..500

M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Formula

G.f.: Sum_{j>=1} (Product_{i=1..j-1} (x/(1-x)-x^i)) j x^j (1-x)/(1-2x).

Example

Example: For n=3 the a(3)=2 values are the first 1s in 111 and 12 (the other compositions 21 and 3 do not have any fixed points).

Prog

(PARI) my(N=44, x='x+O('x^N)); Vec( sum(j=1, N, prod(i=1, j-1, (x/(1-x)-x^i) ) *j*x^j * (1-x)/(1-2*x) ) ) \\ Joerg Arndt, Jun 18 2020

Crossrefs

Cf. A099036, A335713, A335714.

Sequence in context: A029863 A091919 A059078 * A356465 A166963 A188476

Adjacent sequences: A335709 A335710 A335711 * A335713 A335714 A335715

Keyword

nonn

Author

Margaret Archibald, Jun 18 2020

Extensions

a(21)-a(34) from Alois P. Heinz, Jun 18 2020

Status

approved