A003513 Number of regular sequences of length n.

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

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more

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)

Examples

			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)
		

References

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

Links

• Marc Davio, Unpublished notes, 1975, from a letter to N. J. A. Sloane sent in May 1975.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

Crossrefs

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

Cf. A126796.

Programs

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

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