A394611 - OEIS

%I #48 Jul 01 2026 20:14:16

%S 0,0,1,2,2,4,3,4,6,6,8,7,6,10,11,6,12,13,10,15,8,14,18,14,20,16,19,13,

%T 13,15,17,21,26,18,22,22,24,27,22,26,30,28,37,20,26,36,27,27,29,29,31,

%U 30,40,36,35,38,31,33,44,43,37,47,33,46,44,48,46,48,50,51,43,45,38,35,46,55,45,43,49,61,49,63,52

%N a(1) = 0. For n > 1, a(n) is the number of ways to write a(n-1) as the sum of two previous terms with distinct positions in the sequence.

%C Conjecture: Many numbers, beginning with 5 and 9, never occur in this sequence (see A397471).

%C It follows from the Erdős-Fuchs theorem that the average behavior of a representation function cannot be too close to a linear function, but it is still possible to give an estimate for the growth of this sequence as n/sqrt(2) because if we write g(x) = Sum_{k>=0} x^floor(k/sqrt(2)), then we know that [x^m] (1/2)*(g(x)^2-g(x^2)) will be relatively close to m or even exactly m in many cases. - _Thomas Scheuerle_, Jun 29 2026

%H Sean A. Irvine, <a href="/A394611/b394611.txt">Table of n, a(n) for n = 1..10000</a>

%H Sean A. Irvine, <a href="/A394611/a394611_1.txt.gz">Table of n, a(n) for n = 1..1000000</a> [4.3M, gzip]

%F a(n) = [x^a(n-1)] (1/2)*(g(x)^2-g(x^2)), where g(x) = Sum_{k=1..n-1} x^a(k). - _Thomas Scheuerle_, Jun 29 2026

%e a(2) = 0, since we only have one previous term.

%e a(3) = 1, since a(1) + a(2) = a(2).

%e a(4) = 2, since a(1) + a(3) = a(3) and a(2) + a(3) = a(3).

%o (Python)

%o def sequence(N=100):

%o     a = [0]  # a(1) = 0

%o     for n in range(1, N):

%o         target = a[n - 1]  # a(n)

%o         count = 0

%o         # count pairs (i < j) with i,j in [0..n-1]

%o         for i in range(n):

%o             for j in range(i + 1, n):

%o                 if a[i] + a[j] == target:

%o                     count += 1

%o         a.append(count)

%o     return a

%o seq = sequence(100)

%o print(seq)

%o (Python)

%o from collections import Counter

%o from itertools import islice

%o def A394611_gen(): # generator of terms

%o     alst, an, c = [], 0, Counter()

%o     while True:

%o         an = c[an]

%o         yield an

%o         c.update(an+ai for ai in alst)

%o         alst.append(an)

%o print(list(islice(A394611_gen(), 83))) # _Michael S. Branicky_, Jun 26 2026

%o (PARI)

%o ListA(max_n) = {my(M=Map(), v=[0], c=0, p=0);  while(#v<max_n, c=0; mapisdefined(M, v[#v], &c); v=concat(v, c); for(k=1, #v-1, c=0; p=v[#v]+v[k]; mapisdefined(M, p, &c); mapput(~M, p, c+1))); v} \\ _Thomas Scheuerle_, Jun 29 2026

%Y Cf. A393702, A395907, A397173-A397180, A397471.

%K nonn,new

%O 1,4

%A _Joshua B. Weinstein_, Jun 18 2026