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Irregular triangle T(n,k) read by rows in which row n lists the iterates of the Farkas map (A349407) from 2*n - 1 to 1.
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%I #22 Sep 13 2024 08:11:16

%S 1,3,1,5,3,1,7,11,17,9,3,1,9,3,1,11,17,9,3,1,13,7,11,17,9,3,1,15,5,3,

%T 1,17,9,3,1,19,29,15,5,3,1,21,7,11,17,9,3,1,23,35,53,27,9,3,1,25,13,7,

%U 11,17,9,3,1,27,9,3,1,29,15,5,3,1

%N Irregular triangle T(n,k) read by rows in which row n lists the iterates of the Farkas map (A349407) from 2*n - 1 to 1.

%H Paolo Xausa, <a href="/A350279/b350279.txt">Table of n, a(n) for n = 1..12301</a> (rows 1..1000 of triangle, flattened).

%H H. M. Farkas, "Variants of the 3N+1 Conjecture and Multiplicative Semigroups", in Entov, Pinchover and Sageev, <a href="https://bookstore.ams.org/conm-387">Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Mathematics, vol. 387</a>, American Mathematical Society, 2005, p. 121.

%H J. C. Lagarias, ed., <a href="http://www.ams.org/bookstore-getitem/item=mbk-78">The Ultimate Challenge: The 3x+1 Problem</a>, American Mathematical Society, 2010, p. 74.

%F T(n,1) = 2*n-1; T(n,k) = A349407((T(n,k-1)+1)/2), where n >= 1 and k >= 2.

%e Written as an irregular triangle, the sequence begins:

%e n\k 1 2 3 4 5 6 7

%e -------------------------------

%e 1: 1

%e 2: 3 1

%e 3: 5 3 1

%e 4: 7 11 17 9 3 1

%e 5: 9 3 1

%e 6: 11 17 9 3 1

%e 7: 13 7 11 17 9 3 1

%e 8: 15 5 3 1

%e 9: 17 9 3 1

%e 10: 19 29 15 5 3 1

%e 11: 21 7 11 17 9 3 1

%e 12: 23 35 53 27 9 3 1

%t FarkasStep[x_] := Which[Divisible[x, 3], x/3, Mod[x, 4] == 3, (3*x + 1)/2, True, (x + 1)/2];

%t Array[Most[FixedPointList[FarkasStep, 2*# - 1]] &, 15] (* _Paolo Xausa_, Sep 03 2024 *)

%Y Cf. A349407, A375909 (# of iterations), A375910 (row sums), A375911 (row maxs).

%Y Cf. A070165.

%K nonn,easy,tabf

%O 1,2

%A _Paolo Xausa_, Dec 22 2021