A376635 a(n+1) = size of the largest subset S of 1...n such that i+j <= n implies a(i+j) = a(i)+a(j) for i and j in S. Start with a(1) = 1.

1, 1, 1, 2, 3, 3, 4, 5, 4, 5, 5, 6, 7, 8, 9, 9, 8, 9, 10, 11, 12, 13, 12, 12, 13, 14, 15, 16, 17, 17, 16, 17, 17, 17, 17, 18, 19, 20, 21, 22, 23, 23, 22, 23, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 37, 37, 37, 38, 38, 39, 39

Offset: 1

Author

Bryle Morga, Sep 30 2024

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..600

Programs

Maple

f:= proc(n) local V,E,G,i,j; uses GraphTheory;
      V:= select(t -> 2*t > n or 2*A[t] = A[2*t], [$1..n]);
      E:= select(t -> t[1]+t[2] <= n and A[t[1]]+A[t[2]] <> A[t[1]+t[2]],{seq(seq({V[i],V[j]},i=1..j-1),j=1..nops(V))});
      G:= Graph(V,E);
      IndependenceNumber(G)
end proc:
A[1]:= 1:
for n from 1 to 99 do A[n+1]:= f(n) od:
seq(A[i],i=1..100); # Robert Israel, Oct 31 2024

Python

from itertools import combinations, count, islice
def c(n, s, a): # test the condition for subset s
    for ii, i in enumerate(s):
        for j in s[ii:]:
            if i+j <= n:
                if a[i] + a[j] != a[i+j]:
                    return False
            else:
                break
    return True
def agen(): # generator of terms
    a, valid = [None, 1], [tuple()]
    yield 1
    for n in count(1):
        new_valid, r = [], 0
        for s in valid:
            if c(n, s, a):
                new_valid.extend([s, s+(n,)])
                r = max(r, len(s)+1)
        valid = new_valid
        yield r
        a.append(r)
print(list(islice(agen(), 30))) # Michael S. Branicky, Oct 01 2024

Extensions

a(23)-a(58) from Michael S. Branicky, Oct 01 2024

More terms from Robert Israel, Oct 31 2024