A350102 -id:A350102 - cp's OEIS frontend

A350104 a(n) = Sum_{k=0..n} A350102(k).

1, 3, 6, 11, 18, 28, 40, 56, 74, 96, 121, 150, 181, 218, 257, 300, 347, 399, 453, 513, 575, 643, 715, 791, 869, 955, 1044, 1137, 1234, 1337, 1442, 1555, 1670, 1791, 1916, 2045, 2178, 2320, 2464, 2612, 2764, 2924, 3086, 3256, 3428, 3606, 3790, 3978, 4168, 4368

Offset: 0

Peter Luschny, Dec 16 2021

nonn

Cf. A350102, A350103, A350105, A349976.

def A350104List(len):
    L = [1] * len
    a, b = 1, 2
    for n in (2..len):
        a += b
        b += sloane.A000005(n - 1)
        L[n - 1] = a
    return L
print(A350104List(50))

A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 7, 3, 2, 1, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Dec 14 2021

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).

			Triangle starts:
[ 0] [1]
[ 1] [1, 1]
[ 2] [1, 1,  1]
[ 3] [1, 1,  2, 1]
[ 4] [1, 1,  3, 1, 1]
[ 5] [1, 1,  4, 2, 1, 1]
[ 6] [1, 1,  5, 2, 1, 1, 1]
[ 7] [1, 1,  6, 3, 2, 1, 1, 1]
[ 8] [1, 1,  7, 3, 2, 1, 1, 1, 1]
[ 9] [1, 1,  8, 4, 2, 2, 1, 1, 1, 1]
[10] [1, 1,  9, 4, 3, 2, 1, 1, 1, 1, 1]
[11] [1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1]
[12] [1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1]
.
The first  column is 1,1,...  because {} = distset({}) and |{}| = 0.
The second column is 1,1,... because {0} = distset({0}) and |{0}| = 1.
The third  column is n-1  because {0, j} = distset({0, j}) and |{0, j}| = 2 for j = 1..n - 1.
The main diagonal is 1,1,... because [n] = distset([n]) and |[n]| = n (these are the complete rulers A103295).

Winston de Greef, Table of n, a(n) of the first 150 rows, flattened (n = 0..11324)
Peter Luschny, Illustrating self-measuring subsets of {0, 1, 2, 3}.

Cf. A350102 (row sums), A349976, A103295, A003988, A010766, A123706.

T := (n, k) -> ifelse(k < 2, 1, floor((n - 1) / (k - 1))):
seq(print(seq(T(n, k), k = 0..n)), n = 0..12);

distSet[s_] := Union[Map[Abs[Differences[#][[1]]] &, Union[Sort /@ Tuples[s, 2]]]]; T[n_, k_] := Count[Subsets[Range[0, n - 1]], ?((ds = distSet[#]) == # && Length[ds] == k &)]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Amiram Eldar, Dec 16 2021 *)

T(n, k) = if(k<=1, 1, (n - 1) \ (k - 1)) \\ Winston de Greef, Jan 31 2024

# generating and counting (slow)
def isSelfMeasuring(R):
    S, L = Set([]), len(R)
    R = Set([r - 1 for r in R])
    for i in range(L):
        for j in (0..i):
            S = S.union(Set([abs(R[i] - R[i - j])]))
    return R == S
def A349976row(n):
    counter = [0] * (n + 1)
    for S in Subsets(n):
        if isSelfMeasuring(S): counter[len(S)] += 1
    return counter
for n in range(10): print(A349976row(n))

T(n, k) = floor((n - 1) / (k - 1)) for k >= 2.

T(n, k) = 1 if k = 0 or k = 1 or n >= k >= floor((n + 1)/2).

A350105 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).

Original entry on oeis.org

0, 0, 1, 3, 9, 22, 52, 112, 238, 490, 999, 2019, 4065, 8155, 16345, 32725, 65489, 131020, 262090, 524228, 1048514, 2097084, 4194232, 8388532, 16777138, 33554346, 67108775, 134217635, 268435359, 536870809, 1073741719, 2147483535, 4294967181, 8589934471, 17179869059

Offset: 0

Peter Luschny, Dec 16 2021

nonn

Winston de Greef, Table of n, a(n) for n = 0..3305

Cf. A350102, A350103, A349976.

def A350105List(len):
    L = [0] * len
    b, z = 2, 2
    for n in (2..len-1):
        b += sloane.A000005(n - 1)
        z += z
        L[n] = z - b
    return L
print(A350105List(35))

See the formulas in A350102.

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

cp's OEIS Frontend

A350104 a(n) = Sum_{k=0..n} A350102(k).

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A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

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A350105 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).

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