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 A239124 #26 Apr 04 2025 07:59:19
%S A239124 53,117,181,245,309,373,437,501,565,629,693,757,821,885,949,1013,1077,
%T A239124 1141,1205,1269,1333,1397,1461,1525,1589,1653,1717,1781,1845,1909,
%U A239124 1973,2037,2101,2165,2229,2293,2357,2421,2485,2549,2613,2677,2741,2805,2869,2933,2997
%N A239124 a(n) = 64*n - 11 for n >= 1. Third column of triangle A238476.
%C A239124 This sequence gives all start numbers a(n) (sorted increasingly) of Collatz sequences of length 7 following the pattern ud^5 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, requiring that the sequence ends in an odd number. The last entry of this Collatz sequence is 6*n - 1.
%C A239124 This appears in Example 2.1. for x = 5 in the M. Trümper paper given as a link below.
%H A239124 Vincenzo Librandi, <a href="/A239124/b239124.txt">Table of n, a(n) for n = 1..1000</a>
%H A239124 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 A239124 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 A239124 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A239124 O.g.f.: x*(53+11*x)/(1-x)^2.
%F A239124 From _Elmo R. Oliveira_, Apr 04 2025: (Start)
%F A239124 E.g.f.: 11 + exp(x)*(64*x - 11).
%F A239124 a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)
%e A239124 a(1) = 53 because the Collatz sequence of length 7 following the pattern uddddd, ending in an odd number is [53, 160, 80, 40, 20, 10, 5]. The end number is 6*1 - 1 = 5.
%t A239124 CoefficientList[Series[(53 + 11 x)/(1 - x)^2, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 13 2014 *)
%t A239124 64 Range[40]-11 (* _Harvey P. Dale_, Nov 21 2018 *)
%o A239124 (Magma) [64*n-11: n in [1..50]]; // _Vincenzo Librandi_, Mar 13 2014
%Y A239124 Cf. A004767 (first column), A082285 (second column), A238476.
%K A239124 nonn,easy
%O A239124 1,1
%A A239124 _Wolfdieter Lang_, Mar 10 2014