A384468 a(0) = 1; for n >= 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = 2*a(n-1) + n.

1, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 19, 52, 26, 13, 43, 104, 52, 26, 13, 48, 24, 12, 6, 3, 33, 94, 47, 124, 62, 31, 95, 224, 112, 56, 28, 14, 7, 54, 27, 96, 48, 24, 12, 6, 3, 54, 27, 104, 52, 26, 13, 80, 40, 20, 10, 5, 69, 198, 99, 260, 130, 65, 195, 456, 228, 114, 57, 184

Offset: 0

Author

Simon R Blow, May 30 2025

Keywords

• nonn
• easy

Comments

The sequence behaves similarly to the Collatz sequence but introduces a linearly increasing term n when updating odd values.

a(n) = 1 for n = 0, 5, 9, 60843, 19628571, and 772944372 and no other values up to 10^14. - David Radcliffe, Jun 18 2025

Examples

			a(2) = 2*3 + 2 = 8 (since a(1) is odd).
a(3) = 8/2 = 4 (since a(2) is even).
		

Crossrefs

Cf. A000079, A006370.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = If[EvenQ[a[n-1]], a[n-1]/2, 2*a[n-1] + n]; Array[a, 100, 0] (* Amiram Eldar, May 30 2025 *)

Python

def generate_sequence(n_terms):
    seq = [1]  # a(0) = 1
    for n in range(1, n_terms):
        prev = seq[-1]
        if prev % 2 == 0:
            seq.append(prev // 2)
        else:
            seq.append(2 * prev + n)
    return seq
seq = generate_sequence(1000)
print(seq)

Formula

a(0) = 1.

For n >= 1:

a(n) = a(n-1)/2 if a(n-1) is even;

a(n) = 2*a(n-1) + n if a(n-1) is odd.