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Number of subsets of the initial segment of the natural numbers strictly below n which are not self-measuring. Number of subsets S of [n] with S != distset(S).
2

%I #12 Feb 01 2024 08:39:04

%S 0,0,1,3,9,22,52,112,238,490,999,2019,4065,8155,16345,32725,65489,

%T 131020,262090,524228,1048514,2097084,4194232,8388532,16777138,

%U 33554346,67108775,134217635,268435359,536870809,1073741719,2147483535,4294967181,8589934471,17179869059

%N Number of subsets of the initial segment of the natural numbers strictly below n which are not self-measuring. Number of subsets S of [n] with S != distset(S).

%C We use the notation [n] = {0, 1, ..., n-1}. If S is a subset of [n] then we define the distset of S (set of distances of S) as {|x - y|: x, y in S}. We call a subset S of the natural numbers self-measuring if and only if S = distset(S).

%H Winston de Greef, <a href="/A350105/b350105.txt">Table of n, a(n) for n = 0..3305</a>

%F See the formulas in A350102.

%F a(n) = 2^n - A350102(n).

%o (SageMath)

%o def A350105List(len):

%o L = [0] * len

%o b, z = 2, 2

%o for n in (2..len-1):

%o b += sloane.A000005(n - 1)

%o z += z

%o L[n] = z - b

%o return L

%o print(A350105List(35))

%Y Cf. A350102, A350103, A349976.

%K nonn

%O 0,4

%A _Peter Luschny_, Dec 16 2021