A335714 -id:A335714 - cp's OEIS frontend

A335713 The sum of the sizes of the largest fixed points over all compositions of n.

1, 1, 3, 7, 16, 34, 73, 155, 324, 674, 1393, 2861, 5852, 11929, 24239, 49127, 99360, 200598, 404377, 814135, 1637363, 3290067, 6605980, 13255451, 26583994, 53290694, 106787166, 213919062, 428415074, 857794856, 1717201360, 3437092882, 6878672565, 13764822699

Offset: 1

Margaret Archibald, Jun 18 2020

nonn

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

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

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.

Cf. A099036, A335712, A335714.

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

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

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

Original entry on oeis.org

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

Margaret Archibald, Jun 18 2020

nonn

			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).

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

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.

Cf. A099036, A335713, A335714.

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

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

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

cp's OEIS Frontend

A335713 The sum of the sizes of the largest fixed points over all compositions of n.

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