|
%I #12 Jan 01 2022 21:32:50
%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 2n-1 to 1.
%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) = 2n-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 f[x_]:=If[Mod[x,3]==0,x/3,If[Mod[x,4]==3,(3x+1)/2,(x+1)/2]]
%t nrows=15;Table[NestWhileList[f,2n-1,#>1&],{n,nrows}]
%Y Cf. A070165, A349407.
%K nonn,easy,tabf
%O 1,2
%A _Paolo Xausa_, Dec 22 2021
|
