A394252 - OEIS

%I #16 Apr 03 2026 21:51:24

%S 1,2,5,10,28,56,112,326,652,1304,2608,7712,15424,30848,61696,182480,

%T 364960,729920,1459840,2919680,8697344,17394688,34789376,69578752,

%U 139157504,414552832,829105664,1658211328,3316422656,6632845312,13265690624,39657914368

%N a(1)=1; for n>1, a(n) is the smallest integer greater than a(n-1) that cannot be written as the sum of exactly n-1 elements from {a(1),...,a(n-1)}, repetition allowed.

%C Among the first 32 computed terms several consecutive doubling blocks occur, for example 1,2; 5,10; 28,56,112; 326,652,1304,2608; 7712,15424,30848,61696; 182480,364960,729920,1459840,2919680. The lengths of these doubling blocks appear to increase over time (2,2,3,4,4,5,...), and the number of blocks of a given length also appears to increase, although with irregular fluctuations. No proof of these structural properties is known to the author.

%C Empirical observation: a(n+1) appears to be either 2*a(n), or of the form a(n+1) = 3*a(n) - k, where k is the last term of the previous doubling block. No proof of this phenomenon is known to the author.

%e a(1)=1.

%e For n=2, we must use exactly 1 earlier term {1}.

%e Representable numbers: 1.

%e The smallest integer >1 not representable is 2, so a(2)=2.

%e For n=3, we use exactly 2 terms from {1,2}.

%e Representable numbers: 2,3,4.

%e The smallest integer >2 not representable is 5, so a(3)=5.

%e For n=4, we use exactly 3 terms from {1,2,5}.

%e Representable numbers: 6,7,8,9.

%e The smallest integer >5 not representable is 10, so a(4)=10.

%o (Python)

%o from itertools import product

%o def representable(x,t,A):

%o     for c in product(A, repeat=t):

%o         if sum(c)==x:

%o             return True

%o     return False

%o def seq(n):

%o     A=[1]

%o     while len(A)<n:

%o         k=A[-1]+1

%o         while representable(k,len(A),A):

%o             k+=1

%o         A.append(k)

%o     return A

%K nonn

%O 1,2

%A _Haoqian Wen_, Mar 13 2026