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 A238475 #32 Feb 16 2025 08:33:21 %S A238475 1,9,5,17,37,21,25,69,149,85,33,101,277,597,341,41,133,405,1109,2389, %T A238475 1365,49,165,533,1621,4437,9557,5461,57,197,661,2133,6485,17749,38229, %U A238475 21845,65,229,789,2645,8533,25941,70997,152917,87381 %N A238475 Rectangular array with all start numbers Me(n, k), k >= 1, for the Collatz operation ud^(2*n), n >= 1, ending in an odd number, read by antidiagonals. %C A238475 The two operations on natural numbers m used in the Collatz 3x+1 conjecture (see the links) are here (following the M. Trümper reference) 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 positive start numbers Me(n, k), k >= 1, for Collatz sequences following the pattern (word) ud^(2*n), for n >= 1, which end in an odd number. The end number does not depend on n and it is given by Ne(k) = 6*k - 5. %C A238475 This rectangular array is Example 2.1. with x = 2*n, n >= 1, of the M. Trümper reference, pp. 4-5, written as a triangle by taking NE-SW diagonals. The case x = 2*n+1, n >= 0, for the word ud^(2*k+1) appears as array and triangle in A238476. %C A238475 The first row sequences of the array Me (they become columns in the triangle Te) are A017077, A238477, A239123, ... %C A238475 Note that there are also Collatz sequences starting with an odd number, following the pattern ud^(2*n) which end in an even number. For example, take n=1 and the sequence [5, 16, 8, 4]. Such sequences are here not considered. %H A238475 W. 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 A238475 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 A238475 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>. %H A238475 Wikipedia, <a href="https://en.wikipedia.org/wiki/Collatz_conjecture">Collatz Conjecture</a>. %F A238475 The array: Me(n, k) = 2^(2*n+1)*k - (5*2^(2*n)+1)/3 for n >= 1 and k >= 1. %F A238475 The triangle: Te(m, n) = Me(n, m-n+1) = 2*4^n*(m-n) + (4^n-1)/3 for m >= n >= 1 and 0 for m < n. %e A238475 The rectangular array Me(n, k) begins: %e A238475 n\k 1 2 3 4 5 6 7 8 9 10 ... %e A238475 1: 1 9 17 25 33 41 49 57 65 73 %e A238475 2: 5 37 69 101 133 165 197 229 261 293 %e A238475 3: 21 149 277 405 533 661 789 917 1045 1173 %e A238475 4: 85 597 1109 1621 2133 2645 3157 3669 4181 4693 %e A238475 5: 341 2389 4437 6485 8533 10581 12629 14677 16725 18773 %e A238475 6: 1365 9557 17749 25941 34133 42325 50517 58709 66901 75093 %e A238475 7: 5461 38229 70997 103765 136533 169301 202069 234837 267605 300373 %e A238475 8: 21845 152917 283989 415061 546133 677205 808277 939349 1070421 1201493 %e A238475 9: 87381 611669 1135957 1660245 2184533 2708821 3233109 3757397 4281685 4805973 %e A238475 10: 349525 2446677 4543829 6640981 8738133 10835285 12932437 15029589 17126741 19223893 %e A238475 ... %e A238475 The triangle Te(m, n) begins (zeros are not shown): %e A238475 m\n 1 2 3 4 5 6 7 8 9 10 ... %e A238475 1: 1 %e A238475 2: 9 5 %e A238475 3: 17 37 21 %e A238475 4: 25 69 149 85 %e A238475 5: 33 101 277 597 341 %e A238475 6: 41 133 405 1109 2389 1365 %e A238475 7: 49 165 533 1621 4437 9557 5461 %e A238475 8: 57 197 661 2133 6485 17749 38229 21845 %e A238475 9: 65 229 789 2645 8533 25941 70997 152917 87381 %e A238475 10: 73 261 917 3157 10581 34133 103765 283989 611669 349525 %e A238475 ... %e A238475 ---------------------------------------------------------------------------------------------- %e A238475 n=1, ud^2, k=1: Me(1, 1) = 1 = Te(1, 1), Ne(1) = 1 with the Collatz sequence [1, 4, 2, 1] of length 4. %e A238475 n=1, ud^2, k=2: Me(1, 2) = 9 = Te(2, 1), Ne(2) = 7 with the Collatz sequence [9, 28, 14, 7] of length 4. %e A238475 n=2, ud^4, k=1: Me(2, 1) = 5 = Te(2, 2), Ne(1) = 1 with the length 6 Collatz sequence [5, 16, 8, 4, 2, 1]. %e A238475 n=5, ud^(10), k=2: Me(5, 2) = 2389 = Te(6,5), Ne(2) = 7 with the Collatz sequence [2389, 7168, 3584, 1792, 896, 448, 224, 112, 56, 28, 14, 7] of length 12. %Y A238475 Cf. A006577, A139399, A112695, A238476, A017077, A238477, A239123. %K A238475 nonn,easy,tabl %O A238475 1,2 %A A238475 _Wolfdieter Lang_, Mar 10 2014