A383131 -id:A383131 - cp's OEIS frontend

A381055 a(n) = -n/2 if n is even, 3n + 1 if n is odd.

0, 4, -1, 10, -2, 16, -3, 22, -4, 28, -5, 34, -6, 40, -7, 46, -8, 52, -9, 58, -10, 64, -11, 70, -12, 76, -13, 82, -14, 88, -15, 94, -16, 100, -17, 106, -18, 112, -19, 118, -20, 124, -21, 130, -22, 136, -23, 142, -24, 148, -25, 154, -26, 160, -27, 166, -28, 172

Offset: 0

Ya-Ping Lu, Apr 14 2025

If negative indices are allowed, integers m = 0, 1, 2, 3, or 5 (mod 6) appear once (at n = -2*m) in the sequence, while integers m = 4 (mod 6) appear twice (at n = -2*m and (m-1)/3).

Conjecture: Iterating the map x -> a(x) on any negative or positive integer eventually reaches 1. Some iteration paths between -35 and 35 are illustrated in the plot of a(n) in LINKS.

			a(0) = -0/2 = 0; a(1) = 3*1 + 1 = 4; a(10) = -10/2 = -5.

Ya-Ping Lu, A plot of a(n)
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A006370, A383131.

a[n_]:=If[EvenQ[n],-n/2,3n+1];Array[a,58,0] (* James C. McMahon, Apr 30 2025 *)

def A381055(n): return 3*n+1 if n%2 else -n>>1

a(n) = (-1)^(n+1)*A006370(n).
a(n) = (1/4)*((-1)^(n+1)*(7*n+2)+5*n+2).
G.f.: x*(2*x^2-x+4)/((x-1)^2*(x+1)^2). - Alois P. Heinz, Aug 13 2025

A386885 a(n) is the number of iterations that -n requires to reach 1 under the map x -> -x/2 if x is even, 3x + 1 if x is odd.

Original entry on oeis.org

2, 1, 4, 4, 10, 13, 13, 3, 16, 6, 6, 6, 45, 9, 9, 6, 48, 12, 12, 12, 25, 51, 51, 15, 15, 15, 15, 15, 41, 54, 54, 5, 18, 18, 18, 18, 31, 44, 44, 8, 57, 8, 8, 8, 21, 21, 21, 8, 34, 47, 47, 47, 60, 60, 60, 11, 11, 11, 11, 11, 86, 24, 24, 8, 37, 50, 50, 50, 50, 63

Offset: 1

Ryosuke Miyazawa, Aug 06 2025

nonn

Equivalently, a(n) is the number of iterations that n requires to reach -1 under the map x -> -x/2 if x is even, 3x - 1 if x is odd. This is called the shadow Collatz map in the Ryosuke Miyazawa reference.

Conjecture: Every nonzero integer reaches 1. Verified numerically up to +-5*10^7.

			n = 7 requires a(7) = 13 steps to reach -1: 7 -> 20 -> -10 -> 5 -> 14 -> -7 -> -22 -> 11 -> 32 -> -16 -> 8 -> -4 -> 2 -> -1.

Ryosuke Miyazawa, The Shadow Collatz Conjecture: The Journey of All Integers Toward -1, preprint, Zenodo, (2025).

Cf. A006577, A381055, A383131.

a:= proc(n) option remember; `if`(n=-1, 0,
      1+a(`if`(n::even, -n/2, 3*n-1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 13 2025

a[n_]:=Module[{s = 0, m = n},While[m!=-1,s++;m=If[BitAnd[m,1]==1,3*m -1,-m/2]];s];Array[a,70] (* James C. McMahon, Aug 18 2025 *)

a(n) = if(n==0, oo, my(s=0); while(n!=-1, s++; n=if(bitand(n,1), 3*n-1, -n/2)); s) \\ Andrew Howroyd, Aug 06 2025

a(1) = 2; for n > 1, a(n) = 1 + a(f(n)), where f(n) = -n/2 if n is even, 3n - 1 if n is odd.

cp's OEIS Frontend

A381055 a(n) = -n/2 if n is even, 3n + 1 if n is odd.

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