A377029 a(1) = 0; thereafter in the binary expansion of a(n-1), expand bits: 1->01 and 0->10.

0, 2, 6, 22, 406, 92566, 6818458006, 26055178074437806486, 540213899028732737068658940860686756246, 163551003506862550406254063077517364557434408527734307437037618419534882498966

Offset: 1

Author

Darío Clavijo, Oct 13 2024

Keywords

• nonn
• base
• easy

Comments

All terms are even and leading zeros omitted in the final encoding.

Conversely the opposite mapping of bits: 0->01 and 1->10 is A133468.

The bit length of a(n) is 2^(n-1)+1.

The count of bits set for a(n) is A094373(n).

a(n) = 2 (mod 4) for n > 1.

Also all the terms align bitwise to the right.

The hamming distance of a(n) and a(n+1) is in A000079.

Examples

			For n = 5 a(5) = 406 because:
This encoding results in the following tree:
n | a(n)
--+---------------
1 | 0
  | |\
2 | 1 0
  | | |
3 | 1 10
  | | | \
4 | 1 01 10--
  | | |\  \  \
  | | | \  \  \
5 | 1 10 01 01 10
Which also aligns bitwise to the right:
n | a(n)
--+-----------
1 |         0
2 |        10
3 |       110
4 |     10110
5 | 110010110
And 110010110 in base 10 is 406.
		

Links

• Darío Clavijo, Table of n, a(n) for n = 1..13

Crossrefs

Cf. A000051, A000079, A000120, A094373, A133468, A320916, A374625.

Programs

Mathematica

NestList[FromDigits[2 - IntegerDigits[#, 2], 4] &, 0, 10] (* Paolo Xausa, Nov 04 2024 *)

Python

from functools import cache
A374625 = lambda n: int(bin(n)[2:].replace('0', '2'), 4)
@cache
def a(n):
  if n == 1: return 0
  return A374625(a(n-1))
print([a(n) for n in range(1, 12)])

Formula

a(n) = A320916(2^(n-2)+1) for n > 1.

A000120(a(n+1) XOR a(n)) = A000079(n-2).

a(n) = A374625(a(n-1)) for n > 1. - Paolo Xausa, Nov 04 2024