A375156 In the binary expansion of n: expand bits 1 -> 01 and 0 -> x0 from most to least significant, where x is the complement of the previous bit from n.

2, 1, 4, 5, 18, 17, 20, 21, 74, 73, 68, 69, 82, 81, 84, 85, 298, 297, 292, 293, 274, 273, 276, 277, 330, 329, 324, 325, 338, 337, 340, 341, 1194, 1193, 1188, 1189, 1170, 1169, 1172, 1173, 1098, 1097, 1092, 1093, 1106, 1105, 1108, 1109, 1322, 1321, 1316, 1317, 1298, 1297, 1300

Offset: 0

Author

Darío Clavijo, Aug 01 2024

Keywords

• nonn
• base

Comments

At the first bit of n, the previous bit is considered to be 0.

n=0 is a single 0 bit.

Equivalently for n >= 1, between adjacent bits u,v of n, insert bit u NOR v.

This is essentially the MFM(1,3) RLL binary encoding, without leading zeros.

Equivalently, apply (from left to right, in the given order) the following transformations to the binary expansion of n: 10 -> 0100, 1 -> 01, 0 -> 10. - Paolo Xausa, Aug 19 2024

Examples

			For n=137,
  n    = binary  1  0  0  0  1  0  0  1
  a(n) = binary 01 00 10 10 01 00 10 01 = 19017
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Wikipedia, Run-length limited.

Crossrefs

Cf. A008836, A057077, A374625.

Programs

Mathematica

A375156[n_] := FromDigits[StringReplace[IntegerString[n, 2], {"10" -> "0100", "1" -> "01", "0" -> "10"}], 2]; Array[A375156,100,0] (* Paolo Xausa, Aug 19 2024 *)

PARI

mfm_encode(data)=prev_enc_bit=0;enc=[];for(i=1,#data,enc=concat(enc,(1-data[i])*(1-prev_enc_bit));enc=concat(enc,data[i]);prev_enc_bit=data[i];);enc;
a(n)=if(n==0,return(2));fromdigits(mfm_encode(binary(n)),2);
vector(55,n,a(n-1))

Python

def mfm_encode(data):
    prev_enc_bit, enc = "0", ""
    for i in range(len(data)):
        enc += ("1" if data[i] == "0" and prev_enc_bit == "0" else "0")
        enc += data[i]
        prev_enc_bit = enc[-1]
    return enc
def a(n):
    enc = mfm_encode(bin(n)[2:])
    return int("".join(map(str, enc)), 2)
print([a(n) for n in range(0, 55)])

Formula

a(n) = a(n-1) - (-1)^floor(n/2) for n odd.

a(n) mod 2 = 0 for n even.

a(n) = A374625(n) AND a(n) where AND is the bitwise-and.

a(n) OR A374625(n) = A374625(n) where OR is the bitwise-or.