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 A238477 #30 Apr 04 2025 07:59:42
%S A238477 5,37,69,101,133,165,197,229,261,293,325,357,389,421,453,485,517,549,
%T A238477 581,613,645,677,709,741,773,805,837,869,901,933,965,997,1029,1061,
%U A238477 1093,1125,1157,1189,1221,1253,1285,1317,1349,1381,1413,1445,1477,1509,1541,1573
%N A238477 a(n) = 32*n - 27 for n >= 1. Second column of triangle A238475.
%C A238477 This sequence gives all start numbers a(n) (sorted increasingly) of Collatz sequences of length 6 following the pattern udddd = ud^4, 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 and it is given by 6*n - 5.
%C A238477 This appears in Example 2.1. for x = 4 in the M. Trümper paper given as a link below.
%H A238477 Vincenzo Librandi, <a href="/A238477/b238477.txt">Table of n, a(n) for n = 1..1000</a>
%H A238477 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 A238477 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 A238477 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A238477 O.g.f.: x*(5+27*x)/(1-x)^2.
%F A238477 From _Elmo R. Oliveira_, Apr 04 2025: (Start)
%F A238477 E.g.f.: 27 + exp(x)*(32*x - 27).
%F A238477 a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)
%e A238477 a(1) = 5 because the Collatz sequence of length 6 is [5, 16, 8, 4, 2, 1], following the pattern udddd, ending in 1, and 5 is the smallest start number following this pattern ending in an odd number.
%e A238477 a(2) = 37 with the length 6 Collatz sequence [37, 112, 56, 28, 14, 7] ending in 12 - 5 = 7, and this is the second smallest start number with this sequence pattern ending in an odd number.
%t A238477 CoefficientList[Series[(5 + 27 x)/(1 - x)^2, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 12 2014 *)
%Y A238477 Cf. A017077 (first column), A238475, A239123 (third column).
%K A238477 nonn,easy
%O A238477 1,1
%A A238477 _Wolfdieter Lang_, Mar 10 2014