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

0,1

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

Links

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

Wikipedia, Run-length limited.

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.

Example

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

Mathematica

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

Prog

(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)])
(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))

Crossrefs

Cf. A008836, A057077, A374625.

Sequence in context: A336398 A209337 A243004 * A137424 A083007 A309845

Adjacent sequences: A375153 A375154 A375155 * A375157 A375158 A375159

Keyword

nonn,base

Author

Darío Clavijo, Aug 01 2024

Status

approved