A350103 -id:A350103 - cp's OEIS frontend

A349976 Triangle read by rows, number of subsets S of [n] with |distset(S)| = k. T(n, k) for 0 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 2, 3, 1, 5, 10, 4, 8, 4, 1, 6, 15, 6, 17, 10, 9, 1, 7, 21, 9, 31, 17, 25, 17, 1, 8, 28, 12, 51, 27, 47, 49, 33, 1, 9, 36, 16, 77, 43, 77, 97, 93, 63, 1, 10, 45, 20, 112, 62, 113, 169, 177, 187, 128

Offset: 0

Peter Luschny, Dec 09 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}.

For instance a subset S of [n] is a complete ruler (A103295) of length n - 1 if and only if distset(S) = [n].

			Triangle starts:
[n\k]  0, 1,  2,  3,  4,  5,  6,  7,  8,  9
-------------------------------------------
[ 0 ]  1;
[ 1 ]  1, 1;
[ 2 ]  1, 2,  1;
[ 3 ]  1, 3,  3,  1;
[ 4 ]  1, 4,  6,  2,  3;
[ 5 ]  1, 5, 10,  4,  8,  4;
[ 6 ]  1, 6, 15,  6, 17, 10,  9;
[ 7 ]  1, 7, 21,  9, 31, 17, 25, 17;
[ 8 ]  1, 8, 28, 12, 51, 27, 47, 49, 33;
[ 9 ]  1, 9, 36, 16, 77, 43, 77, 97, 93, 63;
.
Let S = {0, 3, 6, 7, 8}. Then S is a subset of [9] and distset(S) = [9].
For n = 7 the 9 subsets S with |distset(S)| = 3 are: {1, 2, 3}, {1, 3, 5}, {1, 4, 7}, {2, 3, 4}, {2, 4, 6}, {3, 4, 5}, {3, 5, 7}, {4, 5, 6}, {5, 6, 7}.

Fausto A. C. Cariboni, Rows n = 0..47, flattened (rows n = 0..36 from Peter Luschny)
Scott Harvey-Arnold, Steven J. Miller, and Fei Peng, Distribution of missing differences in diffsets, arXiv:2001.08931 [math.CO], 2020. Also in: Combinatorial and Additive Number Theory IV, Springer 2021. [Table 2 and Table 3, page 276 and 277 in the book.]
Peter Luschny, The first 37 rows, formatted as a triangular array.
Index entries for sequences related to perfect rulers.

Columns: A000012, A000027, A000217, A002620, A349975.
Cf. A000079 (row sums), A103295 (main diagonal), A349972 (subdiagonal).
Cf A349973, A349974, A350103.

distSetSize[s_] := Length @ Union[Map[Abs[Differences[#][[1]]] &, Union[Sort /@ Tuples[s, 2]]]]; T[n_, k_] := Count[Subsets[Range[0, n - 1]], ?(distSetSize[#] == k &)]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar, Dec 12 2021 *)

from collections import Counter
def DistsetLength(R) :
    S, L = Set([]), len(R)
    for r in R:
        for s in R:
            S = S.union(Set([abs(r - s)]))
    return len(S)
def A349976row(n):
    C = Counter(DistsetLength(s) for s in Subsets(n))
    return [C[k] for k in (0..n)]
for n in (0..9): print(A349976row(n))

A350102 Number of self-measuring subsets of the initial segment of the natural numbers strictly below n. Number of subsets S of [n] with S = distset(S).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 12, 16, 18, 22, 25, 29, 31, 37, 39, 43, 47, 52, 54, 60, 62, 68, 72, 76, 78, 86, 89, 93, 97, 103, 105, 113, 115, 121, 125, 129, 133, 142, 144, 148, 152, 160, 162, 170, 172, 178, 184, 188, 190, 200, 203, 209, 213, 219, 221, 229, 233, 241, 245

Offset: 0

Peter Luschny, Dec 14 2021

nonn

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).

			a(0) = 1 = card({}).
a(4) = 7 = card({}, {0}, {0, 1}, {0, 2}, {0, 3}, {0, 1, 2}, {0, 1, 2, 3}).
a(6) = 12 = card({}, {0}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {0, 5}, {0, 1, 2}, {0, 2, 4}, {0, 1, 2, 3}, {0, 1, 2, 3, 4}, {0, 1, 2, 3, 4, 5}).

Winston de Greef, Table of n, a(n) for n = 0..10000
Peter Luschny, Illustrating self-measuring subsets of {0, 1, 2, 3}.

Cf. A350103, A349976, A006218, A054519, A000005.

A350102 := n -> ifelse(n = 0, 1, 2 + add(iquo(n-1, k), k = 1 .. n-1)):
seq(A350102(n), n = 0 .. 58);

a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 1] + DivisorSigma[0, n - 1];
Table[a[n], {n, 0, 58}]

a(n) = a(n - 1) + tau(n - 1) for n >= 2, tau = A000005.
a(n) = 2 + Sum_{k=1..n-1} floor((n - 1)/k) for n >= 1.
a(n) = 2 + A006218(n - 1) for n >= 1.
a(n) = 1 + A054519(n - 1) for n >= 1.
Row sums of A350103.
a(n) >= n + floor(n/2) + floor(n/3).

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).

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

Original entry on oeis.org

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))

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A349976 Triangle read by rows, number of subsets S of [n] with |distset(S)| = k. T(n, k) for 0 <= k <= n.

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A350102 Number of self-measuring subsets of the initial segment of the natural numbers strictly below n. Number of subsets S of [n] with S = distset(S).

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

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