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 A239129 #21 Apr 04 2025 08:12:14
%S A239129 17,35,53,71,89,107,125,143,161,179,197,215,233,251,269,287,305,323,
%T A239129 341,359,377,395,413,431,449,467,485,503,521,539,557,575,593,611,629,
%U A239129 647,665,683,701,719,737,755,773,791,809,827,845,863,881,899,917,935,953,971
%N A239129 a(n) = 18*n - 1, n >= 1, the second column of triangle A239127 related to the Collatz problem.
%C A239129 This sequence gives all ending values a(n) (in increasing order) of Collatz sequences of length 5 following the pattern (ud)^2, 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 sequence is required to be odd. The first entry is also odd and is given by M(2,n) = 8*n-1 from the array A239126.
%C A239129 This appears as N in Example 2.2. for x=y = 2 in the M. Trümper paper on p. 7, given as a link below.
%H A239129 Vincenzo Librandi, <a href="/A239129/b239129.txt">Table of n, a(n) for n = 1..1000</a>
%H A239129 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 A239129 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 A239129 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A239129 a(n) = 18*n - 1 for n >= 1.
%F A239129 O.g.f.: x*(x+17)/(1-x)^2.
%F A239129 From _Elmo R. Oliveira_, Apr 04 2025: (Start)
%F A239129 E.g.f.: exp(x)*(18*x - 1) + 1.
%F A239129 a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)
%e A239129 a(1) = 17 because the Collatz sequence for M(2,1) = 8*1 - 1 = 7 from A239126 is [7, 22, 11, 34, 17] ending in the odd number 17.
%e A239129 a(4) = 71 with the Collatz sequence of length 5 starting with M(2,4) = 31 given by [31, 94, 47, 142, 71], ending in a(4).
%t A239129 CoefficientList[Series[(x + 17)/(1 - x)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 16 2014 *)
%Y A239129 Cf. A016969 (first column), A239126, A239127.
%K A239129 nonn,easy
%O A239129 1,1
%A A239129 _Wolfdieter Lang_, Mar 13 2014