A349976 -id:A349976 - cp's OEIS frontend

A349972 a(n) = A349976(n, n-1) for n >= 1 and a(0) = 0.

0, 1, 2, 3, 2, 8, 10, 25, 49, 93, 187, 377, 747, 1472, 2975, 5911, 11880, 23734, 47474, 94885, 190623, 380805, 763402, 1528095, 3061916, 6125358, 12278446, 24564954, 49200792, 98478615, 197164774, 394536002, 789993459, 1580640910, 3163602123, 6330608624, 12668987317

Offset: 0

Peter Luschny, Dec 12 2021

nonn

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..47

Cf. A349976.

# Uses function DistsetLength from A349976.
from collections import Counter
def A349972(n): return Counter([DistsetLength(s) for s in Subsets(n)])[n-1]
print([A349972(n) for n in range(10)])

A349973 a(n) = Sum_{k=0..n} k*A349976(n, k).

Original entry on oeis.org

0, 1, 4, 12, 34, 89, 226, 554, 1328, 3104, 7158, 16235, 36420, 80797, 177789, 387968, 841696, 1814473, 3892920, 8313588, 17686711, 37486091, 79213997, 166888364, 350720648, 735273025, 1538259673, 3211682091, 6693935172, 13928034709, 28936571039, 60032053097

Offset: 0

Peter Luschny, Dec 12 2021

nonn

Cf. A349976.

A349974 a(n) = 1 + Sum_{k=0..n} (2k - 1)A349976(n, k).

Original entry on oeis.org

0, 1, 5, 17, 53, 147, 389, 981, 2401, 5697, 13293, 30423, 68745, 153403, 339195, 743169, 1617857, 3497875, 7523697, 16102889, 34324847, 72875031, 154233691, 325388121, 684664081, 1436991619, 3009410483, 6289146455, 13119434889, 27319198507, 56799400255, 117916622547

Offset: 0

Peter Luschny, Dec 10 2021, following a suggestion from Alois P. Heinz

nonn

Cf. A349976, A349973.

A103295 Number of complete rulers with length n.

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 17, 33, 63, 128, 248, 495, 988, 1969, 3911, 7857, 15635, 31304, 62732, 125501, 250793, 503203, 1006339, 2014992, 4035985, 8080448, 16169267, 32397761, 64826967, 129774838, 259822143, 520063531, 1040616486, 2083345793, 4168640894, 8342197304, 16694070805, 33404706520, 66832674546, 133736345590

Offset: 0

Peter Luschny, Feb 28 2005

nonn

For definitions, references and links related to complete rulers see A103294.
Also the number of compositions of n whose consecutive subsequence-sums cover an initial interval of the positive integers. For example, (2,3,1) is such a composition because (1), (2), (3), (3,1), (2,3), and (2,3,1) are subsequences with sums covering {1..6}. - Gus Wiseman, May 17 2019
a(n) ~ c*2^n, where 0.2427 < c < 0.2459. - Fei Peng, Oct 17 2019

			a(4) = 4 counts the complete rulers with length 4, {[0,2,3,4],[0,1,3,4],[0,1,2,4],[0,1,2,3,4]}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..49
Scott Harvey-Arnold, Steven J. Miller, and Fei Peng, Distribution of missing differences in diffsets, arXiv:2001.08931 [math.CO], 2020.
Peter Luschny, Perfect rulers
Hugo Pfoertner, Count complete rulers of given length. FORTRAN program.
Index entries for sequences related to perfect rulers.
Gus Wiseman, Illustration of A103295.

Cf. A103300 (Perfect rulers with length n). Main diagonal of A349976.

Cf. A000079, A126796, A143823, A169942, A325676, A325677, A325683, A325684, A325685.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SubsetQ[ReplaceList[#,{_,s__,_}:>Plus[s]],Range[n]]&]],{n,0,15}] (* Gus Wiseman, May 17 2019 *)

a(n) = Sum_{i=0..n} A103294(n, i) = Sum_{i=A103298(n)..n} A103294(n, i).

a(30)-a(36) from Hugo Pfoertner, Mar 17 2005
a(37)-a(38) from Hugo Pfoertner, Dec 10 2021
a(39) from Hugo Pfoertner, Dec 16 2021

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).

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

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).

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).

A349975 Expansion of g.f. (x^4(x^2 + 2x + 3))/((x - 1)^4(x + 1)(x^2 + x + 1)).

Original entry on oeis.org

0, 0, 0, 0, 3, 8, 17, 31, 51, 77, 112, 155, 208, 272, 348, 436, 539, 656, 789, 939, 1107, 1293, 1500, 1727, 1976, 2248, 2544, 2864, 3211, 3584, 3985, 4415, 4875, 5365, 5888, 6443, 7032, 7656, 8316, 9012, 9747, 10520, 11333, 12187, 13083, 14021, 15004, 16031, 17104

Offset: 0

Peter Luschny, Dec 10 2021

Number of subsets of [n] having a distset (set of distances, see definition in A349976) of cardinality 4.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Column k=4 of A349976.

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))

A350324 Missing even distances in full prime rulers, i.e., even numbers k, 0 < k < p-3 for some prime p, such that k is not the difference of two primes less than or equal to p.

Original entry on oeis.org

88, 112, 118, 140, 182, 202, 214, 242, 284, 292, 298, 316, 322, 338, 358, 388, 400, 410, 422, 448, 470, 478, 490, 512, 526, 532, 548, 578, 622, 632, 664, 682, 692, 700, 710, 718, 742, 760, 772, 778, 788, 800, 812, 830, 838, 844, 862, 868, 886, 892, 898, 910, 920, 928, 952, 958, 982, 1000, 1022, 1040, 1052, 1072, 1078, 1108, 1130, 1136, 1142, 1154, 1162, 1172, 1192, 1204

Offset: 1

Rainer Rosenthal, Dec 24 2021

nonn

Inspired by the notion of 'distset' as in A349976, and the general idea of sets of natural numbers as marks of a 'ruler'.

			a(1) = 88 < p - 3 for prime number p = 97, and there are no primes p1, p2 <= p with 88 = p1 - p2.

Cf. A000230, A001223, A071904, A349976.

primedist := n -> {seq(2*j, j = 0..(ithprime(n) - 3)/2)} minus `union`(seq({seq(abs(ithprime(j) - ithprime(k)), k = 1..j)}, j = 1..n)):
`union`(seq(primedist(j), j = 1..200)); # Peter Luschny, Dec 24 2021

genit(maxx=1300)={arr=List();forstep(x=2,maxx,2,q=nextprime(x+2);if(!isprime(q-x),listput(arr,x)));arr;} \\ Bill McEachen, Feb 09 2022

cp's OEIS Frontend

A349972 a(n) = A349976(n, n-1) for n >= 1 and a(0) = 0.

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A349973 a(n) = Sum_{k=0..n} k*A349976(n, k).

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A349974 a(n) = 1 + Sum_{k=0..n} (2*k - 1)*A349976(n, k).

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A103295 Number of complete rulers with length n.

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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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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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A349975 Expansion of g.f. (x^4*(x^2 + 2*x + 3))/((x - 1)^4*(x + 1)*(x^2 + x + 1)).

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

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A349974 a(n) = 1 + Sum_{k=0..n} (2k - 1)A349976(n, k).

A349975 Expansion of g.f. (x^4(x^2 + 2x + 3))/((x - 1)^4(x + 1)(x^2 + x + 1)).