A003513 Number of regular sequences of length n. (Formerly M1685)

1, 2, 6, 27, 192, 2280, 47097, 1735803, 115867758, 14137353466, 3172486137982, 1315460211433262, 1011773137731861712, 1448486351628212391462, 3872217739919424676743213

Offset

2,2

Comments

From Nathaniel Johnston, Jun 29 2023: (Start)

A sequence x_1, ..., x_n is regular if 1 = x_1 <= x_2 <= ... <= x_n and x_j <= Sum_{i=1..j-1} x_i for all j >= 2. It is immediate from this definition that x_2 = 1 and x_j <= 2^(j-2) for all j >= 2.

A sequence x_1, x_2, ..., x_n is regular if and only if (x_2, ..., x_n) is a complete partition of x_2+...+x_n (see A126796 for the definition of a complete partition). As a result, the number of regular sequences with sum equal to n is given by A126796(n-1).

(End)

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=2..16.

Marc Davio, Unpublished notes, 1975, from a letter to N. J. A. Sloane sent in May 1975.

Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)

Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]

Peter C. Fishburn et al., Van Lier Sequences, Discrete Appl. Math. 27 (1990), pp. 209-220.

Nathaniel Johnston and Sarah Plosker, Laplacian {-1,0,1}- and {-1,1}-diagonalizable graphs, arXiv:2308.15611 [math.CO], 2023.

Example

From Nathaniel Johnston, Jun 29 2023: (Start)
When n = 4, there are 6 regular sequences:
1,1,1,1
1,1,1,2
1,1,1,3
1,1,2,2
1,1,2,3
1,1,2,4
(End)

Maple

A003513 := proc() local a, b, n ; a := {[1, 1]} ; n := 3 ; while true do b := {} ; for s in a do subsa := combinat[choose](s) ; for i in subsa do newa := add(k, k=i) ; if newa >= op(-1, s) then b := b union {[op(s), newa]} ; fi ; od; od; print(n, nops(b) ) ; a := b ; n := n+1 ; od; end: A003513() ; # R. J. Mathar, Oct 22 2007

Crossrefs

Sequences in the Fishburn-Roberts (1989) article: A005269, A005268, A234595, A005272, A003513, A008926.

Cf. A126796.

Sequence in context: A118085 A058712 A011834 * A113731 A113676 A183323

Adjacent sequences: A003510 A003511 A003512 * A003514 A003515 A003516

Keyword

nonn,nice,more

Author

N. J. A. Sloane

Extensions

a(9) from R. J. Mathar, Oct 22 2007

a(10) from Sean A. Irvine, Jun 15 2015

a(11)-a(16) from Bert Dobbelaere, Dec 28 2020

Status

approved