A383909 In the base 4 expansion of n, map: 0 -> 20, 1 -> 21, 2 -> 30, 3 -> 31.

8, 9, 12, 13, 152, 153, 156, 157, 200, 201, 204, 205, 216, 217, 220, 221, 2440, 2441, 2444, 2445, 2456, 2457, 2460, 2461, 2504, 2505, 2508, 2509, 2520, 2521, 2524, 2525, 3208, 3209, 3212, 3213, 3224, 3225, 3228, 3229, 3272, 3273, 3276, 3277, 3288, 3289, 3292, 3293

Offset

0,1

Comments

This is the fixed point of the morphism or substitution rules involved in the regular paperfolding sequence and these rules are: map bits: 00 -> 1000, 01 -> 1001, 10 -> 1100, 11 -> 1101.

This sequence contains no fixed points and all terms have an even bit size.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Wikipedia, Regular paperfolding sequence

Formula

a(n+1) = a(n)+1 for n even.

a(2^k) = ((4^k)(95 + 48(-1)^k) - 8)/15.

Example

For n = 216 a(216) = 55752 because:
216 = 3120_4 and
 pos | b4 | enc
-----+----+------------
   0 | 3  | 31
   1 | 1  | 31 21
   2 | 2  | 31 21 30
   3 | 0  | 31 21 30 20
and 31213020_4 is 55752.

Maple

a:= proc(n) option remember; [8, 9, 12, 13]
      [irem(n, 4, 'q')+1]+`if`(n<4, 0, 16*a(q))
    end:
seq(a(n), n=0..47);  # Alois P. Heinz, May 14 2025

Mathematica

A383909[n_] := FromDigits[StringReplace[IntegerString[n, 4], {"0"->"20", "1"->"21", "2"->"30", "3"->"31"}], 4];
Array[A383909, 50, 0] (* Paolo Xausa, May 22 2025 *)

Prog

(Python)
def tobase4(n):
    if n == 0: return [0]
    d = []
    while n > 0:
        d.append(n & 3)
        n >>= 2
    return d[::-1]
def a(n):
    B4 = ['20', '21', '30', '31']
    return int(''.join(B4[b] for b in tobase4(n)), 4)
print([a(n) for n in range(1, 48)])
(Python)
def A383909(n):
    s = ('0' if n.bit_length()&1 else '')+bin(n)[2:] if n else '00'
    return int(''.join(('1000', '1001', '1100', '1101')[int(s[i:i+2], 2)] for i in range(0, len(s), 2)), 2) # Chai Wah Wu, May 22 2025

Crossrefs

Cf. A014577.

Sequence in context: A199635 A131864 A347392 * A179443 A264828 A394710

Adjacent sequences: A383906 A383907 A383908 * A383910 A383911 A383912

Keyword

nonn,base,easy

Author

Darío Clavijo, May 14 2025

Status

approved