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 A350053 #29 Jun 19 2025 00:22:38
%S A350053 3,113,227,7281,14563,466033,932067,29826161,59652323,1908874353,
%T A350053 3817748707,122167958641,244335917283,7818749353073,15637498706147,
%U A350053 500399958596721,1000799917193443,32025597350190193,64051194700380387
%N A350053 a(n) = (2^(3*n + 3 + (-1)^n) - (6 + (-1)^n))/9, for n >= 1.
%C A350053 Labels of nodes at level L = 1 of the Collatz tree with only odd numbers congruent to 1, 3, and 7 modulo 8, named here CToddr.
%C A350053 a(n) is given by the successor of the non-leaf node labels of the (reduced) Collatz tree with odd numbers (named here CTodd) at level 1 given by A198586(n), for n >= 1. See a comment in A347834 for the construction of CTodd. (For all labels of CTodd at level 1 see {A002450(k)}_{k>=2}.) The present sequence gives the labels of the (further) reduced rooted tree CToddr, at level L = 1. Level L = 0 has the root labeled 1, and this node has a directed 1-cycle.
%C A350053 The successor of a node label u of the tree CTodd is given by (4*u - 1)/3 if u == 1 (mod 6), (2*u - 1)/3 if u == 5 (mod 6), and there is no successor if the label u == 3 (mod 6) (a leaf).
%C A350053 This sequence is motivated by a draft of Immo O. Kerner (see A347834 and the link).
%C A350053 Sorted set of all A385109(A198584(i)), i>0 (conjectured but easy to see). - _Ralf Stephan_, Jun 18 2025
%H A350053 Winston de Greef, <a href="/A350053/b350053.txt">Table of n, a(n) for n = 1..1100</a>
%H A350053 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,65,0,-64).
%F A350053 Bisection: a(2*k-1) = (2^(6*k-1) - 5)/9 = A228871(k), a(2*k) = (4^(3*k+2) - 7)/9 = A350054(k), for k >= 1.
%F A350053 a(n) = (2^(3*n+ 2 + b(n)) - (5 + b(n)))/9, with b(n) = 1 + (-1)^n = A010673(n-1), for n >= 1. See the name.
%F A350053 G.f.: Bisection: x*(3 + 32*x)/((1 - x)*(1 - 64*x)) and x*(113 - 64*x)/((1 - x)*(1 - 64*x)).
%F A350053 G.f.: x*(3 + 113*x + 32*x^2 - 64*x^3)/((1 - x^2)*(1 - 64*x^2)).
%t A350053 a[n_] := (2^(3*n + 3 + (-1)^n) - (6 + (-1)^n))/9; Array[a, 20] (* _Amiram Eldar_, Jan 21 2022 *) (* or *)
%t A350053 LinearRecurrence[{0, 65, 0, -64}, {3, 113, 227, 7281}, 20] (* _Georg Fischer_, Sep 30 2022 *)
%o A350053 (PARI) a(n) = (2^(3*n + 3 + (-1)^n))\9 \\ _Winston de Greef_, Jan 28 2024
%Y A350053 Cf. A002450, A198586, A228871, A347834, A350054.
%K A350053 nonn,easy
%O A350053 1,1
%A A350053 _Wolfdieter Lang_, Jan 20 2022