A358406 -id:A358406 - cp's OEIS frontend

A358402 a(1) = 0; for n > 1, let a(n-1) = m; if a(n-1) is the first occurrence of m then a(n) = 0, but if there is a k < n-1 with a(k) = m, a(n) is the minimum of n-1-k and j, where a(j) is the first occurrence of m in the sequence.

0, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 3, 3, 1, 3, 2, 6, 0, 1, 3, 5, 0, 1, 3, 4, 0, 1, 3, 4, 4, 1, 3, 4, 3, 2, 6, 17, 0, 1, 3, 6, 5, 21, 0, 1, 3, 6, 6, 1, 3, 4, 18, 0, 1, 3, 5, 14, 0, 1, 3, 5, 5, 1, 3, 4, 14, 9, 0, 1, 3, 6, 17, 35, 0, 1, 3, 6, 6, 1, 3, 4, 16, 0, 1, 3, 5, 21, 43, 0, 1, 3, 6, 14, 27, 0, 1

Offset: 1

Scott R. Shannon, Nov 14 2022

nonn

This sequence can be considered a variation of Van Eck's sequence, see A181391, but where the sequence forms a loop of numbers, joined at its first and last term before each a(n) is calculated. When a previously unseen number first appears the following term is 0, and as 0 appears as the first term of the sequence, the next term after these 0's will always be 1.

See A358403 for the index where each number first appears.

			a(5) = 1 as a(4) = 0 and 0 appears as the first term of the sequence.
a(6) = 2 as a(5) = 1 and a(3) = 1, these being separated by two terms.
a(17) = 6 as a(16) = 2 and 2 appears as the sixth term of the sequence. Note that the number of terms between the two previous occurrences of 2 is 16 - 6 = 10 which is larger than 6, so 6 is chosen.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Brady Haran and N. J. A. Sloane, Don't Know (the Van Eck Sequence), Numberphile video (2019).

Cf. A181391, A358403, A358405, A358406, A171862, A171863.

nn = 120; q[] = c[] = 0; m = a[1] = 0; Do[If[c[#] == 0, k = 0; c[#] = q[#] = n - 1, k = Min[n - 1 - c[#], q[#]]; c[#] = n - 1] &[m]; a[n] = m = k; If[k == u, While[c[u] > 0, u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Jan 21 2023 *)

from itertools import count, islice
def agen():
    an, first, prev = 0, {0: 1}, {0: 1}
    for n in count(2):
        yield an
        an1 = 0 if first[an] == n-1 else min(n-1-prev[an], first[an])
        if an1 not in first: first[an1] = prev[an1] = n
        prev[an] = n-1
        an = an1
print(list(islice(agen(), 96))) # Michael S. Branicky, Nov 14 2022

A358405 a(1) = 0; for n > 1, let a(n-1) = m; if a(n-1) is the first occurrence of m then a(n) = 0, but if there is a k < n-1 with a(k) = m, a(n) is the maximum of n-1-k and j, where a(j) is the first occurrence of m in the sequence.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 5, 0, 3, 0, 2, 5, 8, 0, 4, 0, 2, 6, 0, 3, 11, 0, 3, 10, 0, 3, 10, 25, 0, 4, 16, 0, 3, 10, 25, 29, 0, 5, 26, 0, 3, 10, 25, 29, 37, 0, 6, 29, 37, 46, 0, 5, 14, 0, 3, 14, 54, 0, 4, 29, 37, 46, 51, 0, 6, 19, 0, 3, 13, 0, 3, 10, 30, 0, 4, 16, 45, 0, 4, 16, 32, 0, 4, 16, 32, 82

Offset: 1

Scott R. Shannon, Nov 14 2022

This sequence uses the same selection rules as A358402 but here the maximum of the two offsets is chosen for a(n). The terms form an unusual pattern for larger values of n; see the linked images.

See A358406 for the index where each number first appears.

			a(5) = 2 as a(4) = 0 and a(2) = 0, these being separated by two terms.
a(8) = 5 as a(7) = 2 and 2 appears as the fifth term of the sequence. Note that the number of terms between the two previous occurrences of 2 is 7 - 5 = 2 which is smaller than 5, so 5 is chosen.

Scott R. Shannon, Table of n, a(n) for n = 1..10000.
Scott R. Shannon, Image of the first 100000 terms.
Scott R. Shannon, Image of the first 1000000 terms.
Scott R. Shannon, Image of the first 10000000 terms.

Cf. A358406, A181391, A358402, A358403, A171862, A171863.

nn = 120; q[] = c[] = 0; m = a[1] = 0; Do[If[c[#] == 0, k = 0; c[#] = q[#] = n - 1, k = Max[n - 1 - c[#], q[#]]; c[#] = n - 1] &[m]; a[n] = m = k; If[k == u, While[c[u] > 0, u++]], {n, 2, nn}]; Array[a, nn]

from itertools import count, islice
def agen():
    an, first, prev = 0, {0: 1}, {0: 1}
    for n in count(2):
        yield an
        an1 = 0 if first[an] == n-1 else max(n-1-prev[an], first[an])
        if an1 not in first: first[an1] = prev[an1] = n
        prev[an] = n-1
        an = an1
print(list(islice(agen(), 87))) # Michael S. Branicky, Nov 14 2022

A358403 The index of A358402 where n first appears, or 0 if n never appears.

Original entry on oeis.org

1, 3, 6, 9, 25, 21, 17, 109, 198, 67, 860, 114, 148, 261, 57, 137, 82, 37, 52, 215, 447, 43, 105, 404, 267, 163, 414, 94, 154, 1292, 184, 716, 142, 669, 243, 73, 1685, 126, 360, 209, 329, 324, 355, 88, 542, 300, 887, 366, 422, 492, 1053, 1754, 256, 941, 946, 711, 679, 178, 283, 3273, 2221, 225

Offset: 0

Scott R. Shannon, Nov 14 2022

nonn

See A358402 for further details.

Cf. A358402, A358405, A358406, A181391.

from itertools import count, islice
def agen():
    k, an, first, prev = 0, 0, {0: 1}, {0: 1}
    for n in count(2):
        while k in first: yield first[k]; k += 1
        an1 = 0 if first[an] == n-1 else min(n-1-prev[an], first[an])
        if an1 not in first: first[an1] = prev[an1] = n
        prev[an] = n-1
        an = an1
print(list(islice(agen(), 62))) # Michael S. Branicky, Nov 14 2022

cp's OEIS Frontend

A358402 a(1) = 0; for n > 1, let a(n-1) = m; if a(n-1) is the first occurrence of m then a(n) = 0, but if there is a k < n-1 with a(k) = m, a(n) is the minimum of n-1-k and j, where a(j) is the first occurrence of m in the sequence.

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A358405 a(1) = 0; for n > 1, let a(n-1) = m; if a(n-1) is the first occurrence of m then a(n) = 0, but if there is a k < n-1 with a(k) = m, a(n) is the maximum of n-1-k and j, where a(j) is the first occurrence of m in the sequence.

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Mathematica

Python

A358403 The index of A358402 where n first appears, or 0 if n never appears.

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Author

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Python