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
Martin Fuller, Table of n, a(n) for n = 2..70
Marc Davio, Unpublished notes, 1975, from a letter to N. J. A. Sloane sent in May 1975.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.
FORMULA
Let f_{n}(z,t) be the number of n-term extensions of a regular sequence which has last element z and total t. f_0(z,t) = 1. f_1 etc can be calculated symbolically from f_{n+1}(z,t) = Sum_{x=z..t} f_{n}(x,t+x). Then a(n) = f_{n-2}(1,2). - Martin Fuller, Nov 25 2025
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
PROG
(Python)
from sympy import *
def a(n):
f = S(1); x, z, t = symbols('x z t')
for _ in range(n-2): f = Sum(f.subs({z:x, t:t+x}), (x, z, t)).doit()
return f.subs({z:1, t:2}) # Martin Fuller, Nov 25 2025
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
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
a(17) onward from Martin Fuller, Nov 25 2025
STATUS
approved