A339695 -id:A339695 - cp's OEIS frontend

A339731 Let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; a(n) is the distance between g_1 and g_n.

0, 1, 2, 2, 3, 3, 4, 4, 3, 4, 5, 4, 5, 5, 4, 5, 6, 5, 6, 6, 5, 6, 7, 6, 7, 6, 7, 7, 8, 8, 8, 7, 6, 7, 8, 7, 8, 7, 6, 7, 8, 8, 9, 8, 8, 8, 9, 9, 9, 8, 7, 8, 9, 9, 9, 8, 7, 8, 9, 10, 10, 9, 10, 10, 10, 10, 10, 9, 8, 9, 10, 10, 10, 9, 10, 11, 11, 11, 11, 10, 10

Offset: 1

Rémy Sigrist, Dec 14 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, PARI program for A339731

See A339695 for a similar sequence.

Cf. A064413, A064664, A339732, A339733.

PARI
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See Links section.
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abs(a(n) - a(k)) <= abs(n-k) for any n, k > 0.

a(n) = A339733(n, 1).

A339696 Let G be the undirected graph with nodes {g_k, k >= 0} such that for any k >= 0, g_k is connected to g_{k+1} and g_{A006068(k)} is connected to g_{A006068(k+1)}; a(n) is the number of nodes at distance n from g_0.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 3, 6, 7, 6, 5, 8, 9, 10, 5, 8, 10, 16, 12, 15, 12, 12, 7, 14, 18, 18, 16, 20, 17, 18, 7, 14, 18, 18, 17, 25, 29, 33, 21, 28, 30, 31, 22, 25, 24, 22, 11, 18, 24, 38, 31, 39, 35, 37, 25, 42, 46, 37, 29, 37, 33, 30, 11, 18, 24, 38, 31, 39, 35

Offset: 0

Rémy Sigrist, Dec 13 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..4000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Rémy Sigrist, PARI program for A339696

Cf. A003188, A006068, A339695, A339697.

A339697 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0; let G be the undirected graph with nodes {g_k, k >= 0} such that for any k >= 0, g_k is connected to g_{k+1} and g_{A006068(k)} is connected to g_{A006068(k+1)}; T(n, k) is the distance between g_n and g_k.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 1, 3, 4, 2, 1, 1, 2, 4, 4, 3, 2, 0, 2, 3, 4, 3, 3, 3, 1, 1, 3, 3, 3, 4, 2, 2, 2, 0, 2, 2, 2, 4, 5, 3, 1, 2, 1, 1, 2, 1, 3, 5, 6, 4, 2, 2, 1, 0, 1, 2, 2, 4, 6, 6, 5, 3, 3, 2, 1, 1, 2, 3, 3, 5, 6, 6, 5, 4, 4, 3, 2, 0, 2, 3, 4, 4, 5, 6

Offset: 0

Rémy Sigrist, Dec 13 2020

			Array T(n, k) begins:
  n\k|  0  1  2  3  4  5  6  7  8  9  10  11  12
  ---+------------------------------------------
    0|  0  1  2  2  3  4  4  3  4  5   6   6   6
    1|  1  0  1  1  2  3  3  2  3  4   5   5   5
    2|  2  1  0  1  2  3  2  1  2  3   4   4   4
    3|  2  1  1  0  1  2  2  2  3  4   5   5   5
    4|  3  2  2  1  0  1  1  2  3  4   5   5   4
    5|  4  3  3  2  1  0  1  2  3  4   5   4   3
    6|  4  3  2  2  1  1  0  1  2  3   4   4   4
    7|  3  2  1  2  2  2  1  0  1  2   3   3   3
    8|  4  3  2  3  3  3  2  1  0  1   2   2   2
    9|  5  4  3  4  4  4  3  2  1  0   1   1   2
   10|  6  5  4  5  5  5  4  3  2  1   0   1   2
   11|  6  5  4  5  5  4  4  3  2  1   1   0   1
   12|  6  5  4  5  4  3  4  3  2  2   2   1   0

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Rémy Sigrist, PARI program for A339697
Rémy Sigrist, Colored representation of the table for 0 <= x, y <= 2^10 (where the hue is function of T(x, y))

Cf. A003188, A006068, A339695, A339696.

PARI
```
See Links section.
```

T(n, n) = 0.
T(n, k) = T(k, n).
T(n, k) <= abs(n-k).
T(m, k) <= T(m, n) + T(n, k).
T(n, 0) = A339695(n).

cp's OEIS Frontend

A339731 Let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; a(n) is the distance between g_1 and g_n.

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A339696 Let G be the undirected graph with nodes {g_k, k >= 0} such that for any k >= 0, g_k is connected to g_{k+1} and g_{A006068(k)} is connected to g_{A006068(k+1)}; a(n) is the number of nodes at distance n from g_0.

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A339697 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0; let G be the undirected graph with nodes {g_k, k >= 0} such that for any k >= 0, g_k is connected to g_{k+1} and g_{A006068(k)} is connected to g_{A006068(k+1)}; T(n, k) is the distance between g_n and g_k.

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Examples

Links

Crossrefs

Programs

PARI

Formula