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 A239126 #40 Feb 16 2025 08:33:21 %S A239126 3,7,7,11,15,15,15,23,31,31,19,31,47,63,63,23,39,63,95,127,127,27,47, %T A239126 79,127,191,255,255,31,55,95,159,255,383,511,511,35,63,111,191,319, %U A239126 511,767,1023,1023,39,71,127,223,383,639,1023,1535,2047,2047 %N A239126 Rectangular array showing the starting values M(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals. %C A239126 The companion array and triangle for the odd end numbers N(n, k) is given in A239127. %C A239126 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 the Collatz word (ud)^n = s^n (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. Such Collatz sequences have the maximal number of u's (grow fastest). %C A239126 This rectangular array is M of Example 2.2. with x=y = n, n >= 1, of the M. Trümper reference, pp. 7-8, written as a triangle by taking NE-SW diagonals. The Collatz sequence starting with M(n, k) has length 2*n+1 for each k and it ends in the odd number N(n, k) given in A239127. %C A239126 The first row sequences of the array M (columns of triangle TM) are A004767, A004771, A125169, A239128, ... %H A239126 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 A239126 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 A239126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>. %H A239126 Wikipedia, <a href="https://en.wikipedia.org/wiki/Collatz_conjecture">Collatz Conjecture</a>. %F A239126 The array: M(n, k) = 2^(n+1)*k - 1 for n >= 1 and k >= 1. %F A239126 The triangle: TM(m, n) = M(n, m-n+1) = 2^(n+1)*(m-n+1) - 1 for m >= n >= 1 and 0 for m < n. %F A239126 a(n) = 4*A087808(A130328(n-1)) - 1 (conjectured). - _Christian Krause_, Jun 15 2021 %e A239126 The rectangular array M(n, k) begins: %e A239126 n\k 1 2 3 4 5 6 7 8 9 10 ... %e A239126 1: 3 7 11 15 19 23 27 31 35 39 %e A239126 2: 7 15 23 31 39 47 55 63 71 79 %e A239126 3: 15 31 47 63 79 95 111 127 143 159 %e A239126 4: 31 63 95 127 159 191 223 255 287 319 %e A239126 5: 63 127 191 255 319 383 447 511 575 639 %e A239126 6: 127 255 383 511 639 767 895 1023 1151 1279 %e A239126 7: 255 511 767 1023 1279 1535 1791 2047 2303 2559 %e A239126 8: 511 1023 1535 2047 2559 3071 3583 4095 4607 5119 %e A239126 9: 1023 2047 3071 4095 5119 6143 7167 8191 9215 10239 %e A239126 10: 2047 4095 6143 8191 10239 12287 14335 16383 18431 20479 %e A239126 ... %e A239126 The triangle TM(m, n) begins (zeros are not shown): %e A239126 m\n 1 2 3 4 5 6 7 8 9 10 ... %e A239126 1: 3 %e A239126 2: 7 7 %e A239126 3: 11 15 15 %e A239126 4: 15 23 31 31 %e A239126 5: 19 31 47 63 63 %e A239126 6: 23 39 63 95 127 127 %e A239126 7: 27 47 79 127 191 255 255 %e A239126 8: 31 55 95 159 255 383 511 511 %e A239126 9: 35 63 111 191 319 511 767 1023 1023 %e A239126 10: 39 71 127 223 383 639 1023 1535 2047 2047 %e A239126 ... %e A239126 --------------------------------------------------------------------- %e A239126 n=1, ud, k=1: M(1, 1) = 3 = TM(1, 1), N(1,1) = 5 with the Collatz sequence [3, 10, 5] of length 3. %e A239126 n=1, ud, k=2: M(1, 2) = 7 = TM(2, 1), N(1,2) = 11 with the Collatz sequence [7, 22, 11] of length 3. %e A239126 n=4, (ud)^4, k=2: M(4, 2) = 63 = TM(5, 4), N(4,2) = 323 with the Collatz sequence [63, 190, 95, 286, 143, 430, 215, 646, 323] of length 9. %e A239126 n=5, (ud)^5, k=1: M(5, 1) = 63 = TM(5, 5), N(5,1) = 485 with the Collatz sequence [63, 190, 95, 286, 143, 430, 215, 646, 323, 970, 485] of length 11. %Y A239126 Cf. A006577, A139399, A112695, A238475, A238476, A004767, A004771, A125169, A239127, A239128. %K A239126 nonn,tabl,easy %O A239126 1,1 %A A239126 _Wolfdieter Lang_, Mar 13 2014