OFFSET
1,3
COMMENTS
From Fishburn et al.'s abstract (from the 1993 article): "Recent research in uniqueness of representability for finite measurement structures has identified a number of novel finite integer sequences. One of the simplest, called an elementary sequence, is a nondecreasing integer sequence x_1,x_2,...,x_n with x_1=x_2=1 and, for all k > 2, if x_k > 1 then x_k=x_i+x_j for distinct i,j < k." - Martin Fuller, Nov 29 2025
REFERENCES
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)
LINKS
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 and Fred S. Roberts, Elementary sequences, sub-Fibonacci sequences, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.
EXAMPLE
There are 4 elementary sequences of length 4: (1,1,1,1) has 1 permutation; (1,1,1,2) has 4 permutations; (1,1,2,2) has 6 permutations; (1,1,2,3) has 12 permutations. Thus a(4)=23 permutations in total. - Martin Fuller, Nov 29 2025
PROG
(Python)
import itertools
import math
from collections import Counter
def a234595(n, x=(1, 1)):
if len(x) >= n:
return multinomial(Counter(x).values())
return sum(a234595(n, x+(y, )) for y in ((1, 2) if x[-1] == 1 else
{a+b for a, b in itertools.combinations(x, 2) if a+b >= x[-1]}))
def multinomial(counts):
return math.prod(map(math.comb, itertools.accumulate(counts), counts))
# Martin Fuller, Dec 01 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 04 2014
EXTENSIONS
a(9)-a(19) from Martin Fuller, Dec 05 2025
STATUS
approved