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 A240222 #28 Feb 16 2025 08:33:21 %S A240222 1,3,1,5,9,1,7,17,33,1,9,25,65,129,1,11,33,97,257,513,1,13,41,129,385, %T A240222 1025,2049,1,15,49,161,513,1537,4097,8193,1,17,57,193,641,2049,6145, %U A240222 16385,32769,1,19,65,225,769,2561,8193,24577,65537,131073,1,21,73,257,897,3073,10241,32769,98305 %N A240222 Rectangular array giving all start values M(n, k), k >= 1, for Collatz sequences following the pattern (udd)^(n-1) ud, n >= 1, read by antidiagonals. %C A240222 The companion array and triangle for the end numbers N(n, k) is given in A240223. %C A240222 The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) denoted by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers M(n, k) for Collatz sequences realizing the Collatz word (udd)^n ud = (sd)^n s (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. The length of these Collatz sequences 3*n. For these Collatz sequences M(n, 0) = M(1, 0) = 1 and N(n, 0) = N(1, 0) = 2. %H A240222 Wolfdieter Lang, <a href="/A240222/a240222.pdf">Rectangular array and triangle.</a> %H A240222 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 A240222 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. %H A240222 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>. %H A240222 Wikipedia, <a href="https://en.wikipedia.org/wiki/Collatz_conjecture">Collatz Conjecture</a> %F A240222 The array: M(n, k) = 1 + 2^(2*n-1)*k for n >= 1 and k >= 0. %F A240222 The triangle: TM(m, n) = M(n,m-n+1) = 1 + 2^(2*n-1)*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n. %e A240222 The rectangular array M(n, k) begins: %e A240222 n\k 0 1 2 3 4 5 ... %e A240222 1: 1 3 5 7 9 11 %e A240222 2: 1 9 17 25 33 41 %e A240222 3: 1 33 65 97 129 161 %e A240222 4: 1 129 257 385 513 641 %e A240222 5: 1 513 1025 1537 2049 2561 %e A240222 6: 1 2049 4097 6145 8193 10241 %e A240222 7: 1 8193 16385 24577 32769 40961 %e A240222 8: 1 32769 65537 98305 131073 163841 %e A240222 9: 1 131073 262145 393217 524289 655361 %e A240222 10: 1 524289 1048577 1572865 2097153 2621441 %e A240222 ... %e A240222 For more columns see the link. %e A240222 The triangle TM(m, n) begins (zeros are not shown): %e A240222 k\n 1 2 3 4 5 6 7 ... %e A240222 0: 1 %e A240222 1: 3 1 %e A240222 2: 5 9 1 %e A240222 3: 7 17 33 1 %e A240222 4: 9 25 65 129 1 %e A240222 5: 11 33 97 257 513 1 %e A240222 6: 13 41 129 385 1025 2049 1 %e A240222 ... %e A240222 For more rows see the link. %e A240222 n=1, ud, k=0: M(1, 0) = 1 = TM(0, 1), N(1, 0) = 2 with the Collatz sequence [1, 4, 2] of %e A240222 length 3. %e A240222 n=1, ud, k=2: M(1, 2) = 5 = TM(2, 1), N(1, 2) = 8 with the Collatz sequence [5, 16, 8] of length 3. %e A240222 n=2, uddud, k=0: M(2, 0) = 1 = TM(1, 2), Ne(2, 0) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9. %Y A240222 Cf. A238475, A238476, A239126, A239127. %K A240222 nonn,easy,tabl %O A240222 1,2 %A A240222 _Wolfdieter Lang_, Apr 02 2014