A324245 The modified Collatz map considered by Vaillant and Delarue.

0, 2, 0, 5, 3, 8, 1, 11, 6, 14, 2, 17, 9, 20, 3, 23, 12, 26, 4, 29, 15, 32, 5, 35, 18, 38, 6, 41, 21, 44, 7, 47, 24, 50, 8, 53, 27, 56, 9, 59, 30, 62, 10, 65, 33, 68, 11, 71, 36, 74, 12, 77, 39, 80, 13, 83, 42, 86, 14, 89, 45, 92, 15, 95, 48, 98, 16, 101, 51, 104, 17, 107, 54, 110, 18, 113, 57, 116, 19, 119, 60

Offset: 0

Author

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

Keywords

• nonn
• easy

Comments

This is a modified Collatz-Terras map (A060322), called in the Vaillant and Delarue link f.

The Collatz conjecture: iterations of the map f = a: N_0 -> N_0 with n -> a(n) lead always to 0.

The minimal number k with a^{[k]}(n) = 0 is given by A324037(n).

The tree CfTree, related to this map, giving the branches which lead to 0 for each vertex label of level n >= 0 is given in A324246.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20000
• Nicolas Vaillant and Philippe Delarue, The hidden face of the 3x+1 problem. Part I: Intrinsic algorithm, April 26 2019.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Crossrefs

Cf. A060322, A324037, A324246, A349414.

Programs

Mathematica

a[n_]:=If[OddQ@n,(3n+1)/2,If[Mod[n,4]==0,3n/4,(n-2)/4]];Array[a,51,0] (* Giorgos Kalogeropoulos, Dec 08 2021 *)

PARI

A324245(n) = if(n%2, (1+3*n)/2, if(!(n%4), 3*(n/4), (n-2)/4)); \\ (After Mathematica-code) - Antti Karttunen, Dec 09 2021

Formula

a(n) = (3*n+1)/2 if n is odd, 3*n/4 if n == 0 (mod 4), and (n-2)/4 if n == 2 (mod 4).

a(n) = A349414(n) + n. - Ruud H.G. van Tol, Dec 08 2021

G.f.: x*(2 + 5*x^2 + 3*x^3 + 4*x^4 + x^5 + x^6)/(1 - x^4)^2. - Stefano Spezia, Dec 08 2021

Extensions

More terms from Antti Karttunen, Dec 09 2021