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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.
3

%I #5 Dec 18 2020 04:03:46

%S 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,

%T 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,

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

%N 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.

%H Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625, Dec 08, 2020.

%H Rémy Sigrist, <a href="/A339697/a339697.gp.txt">PARI program for A339697</a>

%H Rémy Sigrist, <a href="/A339697/a339697.png">Colored representation of the table for 0 <= x, y <= 2^10</a> (where the hue is function of T(x, y))

%F T(n, n) = 0.

%F T(n, k) = T(k, n).

%F T(n, k) <= abs(n-k).

%F T(m, k) <= T(m, n) + T(n, k).

%F T(n, 0) = A339695(n).

%e Array T(n, k) begins:

%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12

%e ---+------------------------------------------

%e 0| 0 1 2 2 3 4 4 3 4 5 6 6 6

%e 1| 1 0 1 1 2 3 3 2 3 4 5 5 5

%e 2| 2 1 0 1 2 3 2 1 2 3 4 4 4

%e 3| 2 1 1 0 1 2 2 2 3 4 5 5 5

%e 4| 3 2 2 1 0 1 1 2 3 4 5 5 4

%e 5| 4 3 3 2 1 0 1 2 3 4 5 4 3

%e 6| 4 3 2 2 1 1 0 1 2 3 4 4 4

%e 7| 3 2 1 2 2 2 1 0 1 2 3 3 3

%e 8| 4 3 2 3 3 3 2 1 0 1 2 2 2

%e 9| 5 4 3 4 4 4 3 2 1 0 1 1 2

%e 10| 6 5 4 5 5 5 4 3 2 1 0 1 2

%e 11| 6 5 4 5 5 4 4 3 2 1 1 0 1

%e 12| 6 5 4 5 4 3 4 3 2 2 2 1 0

%o (PARI) See Links section.

%Y Cf. A003188, A006068, A339695, A339696.

%K nonn,tabl

%O 0,4

%A _Rémy Sigrist_, Dec 13 2020