This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A240223 #27 Oct 20 2023 06:44:32 %S A240223 2,5,2,8,11,2,11,20,29,2,14,29,56,83,2,17,38,83,164,245,2,20,47,110, %T A240223 245,488,731,2,23,56,137,326,731,1460,2189,2,26,65,164,407,974,2189, %U A240223 4376,6563,2,29,74,191,488,1217,2918,6563,13124,19685,2,32,83,218,569,1460,3647,8750,19685,39368,59051,2 %N A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals. %C A240223 The companion array and triangle for the start numbers M(n, k) is given in A240222. %C A240223 For the Collatz operations u (for 'up') and d (for 'down') see the comment on A240222, also for links, especially for the M. Trümper paper. %H A240223 Wolfdieter Lang, <a href="/A240223/a240223.pdf">Rectangular array and triangle.</a> %H A240223 Wolfdieter Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Lang/lang6.html">On Collatz Words, Sequences, and Trees</a>, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7. %H A240223 Manfred Trümper, <a href="http://dx.doi.org/10.1155/2014/756917">The Collatz Problem in the Light of an Infinite Free Semigroup</a>, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages. %F A240223 The array: N(n, k) = 2 + 3^n*k for n >= 1 and k >= 0. %F A240223 The triangle: TN(m, n) = N(n,m-n+1) = 2 + 3^n*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n. %e A240223 The rectangular array N(n, k) begins %e A240223 n\k 0 1 2 3 4 5 ... %e A240223 1: 2 5 8 11 14 17 %e A240223 2: 2 11 20 29 38 47 %e A240223 3: 2 29 56 83 110 137 %e A240223 4: 2 83 164 245 326 407 %e A240223 5: 2 245 488 731 974 1217 %e A240223 6: 2 731 1460 2189 2918 3647 %e A240223 7: 2 2189 4376 6563 8750 10937 %e A240223 8: 2 6563 13124 19685 26246 32807 %e A240223 9: 2 19685 39368 59051 78734 98417 %e A240223 10: 2 59051 118100 177149 236198 295247 %e A240223 ... %e A240223 For more columns see the link. %e A240223 The triangle TN(m, n) begins (zeros are not shown): %e A240223 m\n 1 2 3 4 5 6 7 ... %e A240223 0: 2 %e A240223 1: 5 2 %e A240223 2: 8 11 2 %e A240223 3: 11 20 29 2 %e A240223 4: 14 29 56 83 2 %e A240223 5: 17 38 83 164 245 2 %e A240223 6: 20 47 110 245 488 731 2 %e A240223 ... %e A240223 For more rows see the link. %e A240223 n=1, ud, k=0: M(1, 0) = 1, N(1, 0) = TN(0, 1) = 2 with the Collatz sequence [1, 4, 2] of length 3. %e A240223 n=1, ud, k=2: M(1, 2) = 5, N(1, 2) = TN(2, 1) = 8 with the Collatz sequence [5, 16, 8] of length 3. %e A240223 n=2, uddud, k=0: M(2, 0) = 1, Ne(2, 0) = TN(1, 2) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9. %Y A240223 Cf. A238475, A238476, A239126, A239127, A240222, A016789 (first row of N), A017185 (second row of N). %K A240223 nonn,easy,tabl %O A240223 0,1 %A A240223 _Wolfdieter Lang_, Apr 04 2014