A350053 a(n) = (2^(3*n + 3 + (-1)^n) - (6 + (-1)^n))/9, for n >= 1.

3, 113, 227, 7281, 14563, 466033, 932067, 29826161, 59652323, 1908874353, 3817748707, 122167958641, 244335917283, 7818749353073, 15637498706147, 500399958596721, 1000799917193443, 32025597350190193, 64051194700380387

Offset: 1

Author

Wolfdieter Lang, Jan 20 2022

Keywords

• nonn
• easy

Comments

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.

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.

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).

This sequence is motivated by a draft of Immo O. Kerner (see A347834 and the link).

Sorted set of all A385109(A198584(i)), i>0 (conjectured but easy to see). - Ralf Stephan, Jun 18 2025

Links

• Winston de Greef, Table of n, a(n) for n = 1..1100
• Index entries for linear recurrences with constant coefficients, signature (0,65,0,-64).

Crossrefs

Cf. A002450, A198586, A228871, A347834, A350054.

Programs

Mathematica

a[n_] := (2^(3*n + 3 + (-1)^n) - (6 + (-1)^n))/9; Array[a, 20] (* Amiram Eldar, Jan 21 2022 *) (* or *)
LinearRecurrence[{0, 65, 0, -64}, {3, 113, 227, 7281}, 20] (* Georg Fischer, Sep 30 2022 *)

PARI

a(n) = (2^(3*n + 3 + (-1)^n))\9 \\ Winston de Greef, Jan 28 2024

Formula

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.

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.

G.f.: Bisection: x*(3 + 32*x)/((1 - x)*(1 - 64*x)) and x*(113 - 64*x)/((1 - x)*(1 - 64*x)).

G.f.: x*(3 + 113*x + 32*x^2 - 64*x^3)/((1 - x^2)*(1 - 64*x^2)).