A347913 a(n) is the number of multisets of integers that are possible to reach by starting with n occurrences of 0 and by splitting. Splitting is taking 2 occurrences of the same integer and incrementing one of them by 1 and decrementing the other occurrence by 1.

1, 1, 2, 2, 7, 9, 29, 47, 144, 264, 747, 1531, 4147, 9063, 23744, 54522, 140223, 332033, 845111, 2045007, 5176880, 12713772, 32115727, 79676437, 201227865, 502852973

Offset

0,3

Comments

If the limit of a(n+1)/a(n) exists, then it is contained in the closed interval [2,6.75]. See Links for proof. Reverse splitting is defined in A348532.

Links

Table of n, a(n) for n=0..25.

Tejo Vrush, Limiting ratio for consecutive terms (Upper bound)

Tejo Vrush, Limiting ratio for consecutive terms (Lower bound)

Example

For n = 5, the multisets are as follows:
  {{0,0,0,0,0}}   {{-1,0,0,0,1}}   {{-1,-1,0,1,1}}
  {{-1,-1,0,0,2}} {{-1,-1,-1,1,2}} {{-2,0,0,1,1}}
  {{-2,0,0,0,2}}  {{-2,-1,1,1,1}}  {{-2,-1,0,1,2}}
Therefore, a(5) = 9.
For n = 6, the multisets are as follows:
  {{0,0,0,0,0,0}}    {{-1,0,0,0,0,1}}   {{-1,-1,0,0,1,1}}
  {{-1,-1,0,0,0,2}}  {{-1,-1,-1,1,1,1}} {{-1,-1,-1,0,1,2}}
  {{-2,0,0,0,1,1}}   {{-2,0,0,0,0,2}}   {{-2,-1,0,1,1,1}}
  {{-2,-1,0,0,1,2}}  {{-2,-1,-1,1,1,2}} {{-2,-1,-1,0,2,2}}
  {{-2,-1,-1,0,1,3}} {{-2,-2,0,1,1,2}}  {{-2,-2,0,0,2,2}}
  {{-2,-2,0,0,1,3}}  {{-2,-2,-1,1,2,2}} {{-2,-2,-1,1,1,3}}
  {{-2,-2,-1,0,2,3}} {{-3,-1,0,1,1,2}}  {{-3,-1,0,0,2,2}}
  {{-3,-1,0,0,1,3}}  {{-3,-1,-1,1,2,2}} {{-3,-1,-1,1,1,3}}
  {{-3,-1,-1,0,2,3}} {{-3,-2,0,1,2,2}}  {{-3,-2,0,1,1,3}}
  {{-3,-2,0,0,2,3}}  {{-3,-2,-1,1,2,3}}
Therefore, a(6) = 29.

Maple

b:= proc(p) option remember; {p, seq(`if`(coeff(p, x, i)>1,
      b(expand((p-2*x^i+x^(i-1)+x^(i+1))*`if`(i=0, x, 1)
               )), [])[], i=0..degree(p))}
    end:
a:= n-> nops(b(n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 07 2021

Mathematica

b[p_] := b[p] = Union@Flatten@Join[{p}, Table[If[Coefficient[p, x, i] > 1, b[Expand[(p - 2*x^i + x^(i - 1) + x^(i + 1))*If[i == 0, x, 1]]]], {i, 0, Exponent[p, x]}]];
a[n_] := If[n == 0, 1, Length[b[n]] - 1];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 04 2022, after Alois P. Heinz *)

Prog

(Python)
def nextq(q):
    used = set()
    for i in range(len(q)-1):
        for j in range(i+1, len(q)):
            if q[i] == q[j]:
                if q[i] in used: continue
                used.add(q[i])
                qc = list(q); qc[i] -= 1; qc[j] += 1
                yield tuple(sorted(qc))
def a(n):
    s = tuple(0 for i in range(n)); reach = {s}; expand = list(reach)
    while len(expand) > 0:
        q = expand.pop()
        for qq in nextq(q):
            if qq not in reach:
                reach.add(qq)
                if len(set(qq)) < len(qq):
                    expand.append(qq)
    return len(reach)
print([a(n) for n in range(17)]) # Michael S. Branicky, Oct 10 2021

Crossrefs

Cf. A348532.

Sequence in context: A267214 A107386 A095021 * A348532 A275282 A307633

Adjacent sequences: A347910 A347911 A347912 * A347914 A347915 A347916

Keyword

nonn,more

Author

Tejo Vrush, Oct 07 2021

Extensions

a(15)-a(22) from David A. Corneth, Oct 08 2021

a(23)-a(25) from Michael S. Branicky, Oct 12 2021

Status

approved