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 A239123 #30 Apr 04 2025 07:59:29
%S A239123 21,149,277,405,533,661,789,917,1045,1173,1301,1429,1557,1685,1813,
%T A239123 1941,2069,2197,2325,2453,2581,2709,2837,2965,3093,3221,3349,3477,
%U A239123 3605,3733,3861,3989,4117,4245,4373,4501,4629,4757,4885,5013,5141,5269,5397,5525,5653
%N A239123 a(n) = 128*n - 107 for n >= 1. Third column of triangle A238475.
%C A239123 This sequence gives all start numbers a(n) (sorted increasingly) of Collatz sequences of length 8 following the pattern ud^6 with u (for `up'), mapping an odd number m to 3*m+1, and d (for `down'), mapping an even number m to m/2. The last entry of this Collatz sequence is required to be odd, and it is given by 6*n - 5.
%C A239123 This appears in Example 2.1. for x = 6 in the M. Trümper paper given as a link below.
%H A239123 Vincenzo Librandi, <a href="/A239123/b239123.txt">Table of n, a(n) for n = 1..1000</a>
%H A239123 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 A239123 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 A239123 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A239123 O.g.f.: x*(21+107*x)/(1-x)^2.
%F A239123 From _Elmo R. Oliveira_, Apr 04 2025: (Start)
%F A239123 E.g.f.: 107 + exp(x)*(128*x - 107).
%F A239123 a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)
%e A239123 a(1) = 21 because the Collatz sequence of length 8 is [21, 64, 32, 16, 8, 4, 2, 1] ending in 6*1-5 = 1, and 21 is the smallest positive number following this pattern udddddd ending in an odd number.
%e A239123 a(2) = 149 with the length 8 Collatz sequence [149, 448, 224, 112, 56, 28, 14, 7] ending in 6*2 - 5 = 7, and 149 is the second smallest start number following this pattern ud^6, ending in an odd number.
%t A239123 CoefficientList[Series[(21 + 107 x)/(1 - x)^2, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 12 2014 *)
%Y A239123 Cf. A238475, A238477 (second column).
%K A239123 nonn,easy
%O A239123 1,1
%A A239123 _Wolfdieter Lang_, Mar 10 2014