A320261 Write n in binary, then modify each run of 0's and each run of 1's by prepending a 1. a(n) is the decimal equivalent of the result.

3, 14, 7, 28, 59, 30, 15, 56, 115, 238, 119, 60, 123, 62, 31, 112, 227, 462, 231, 476, 955, 478, 239, 120, 243, 494, 247, 124, 251, 126, 63, 224, 451, 910, 455, 924, 1851, 926, 463, 952, 1907, 3822, 1911, 956, 1915, 958, 479, 240, 483, 974, 487, 988, 1979, 990

Offset

1,1

Comments

A variation of A175046. Indices of record values are given by A319423.

From Chai Wah Wu, Nov 25 2018 : (Start)

Let f(k) = Sum_{i=2^k..2^(k+1)-1} a(i), i.e., the sum ranges over all numbers with a (k+1)-bit binary expansion. Thus f(0) = a(1) = 3 and f(1) = a(2) + a(3) = 21, and f(2) = a(4)+a(5)+a(6)+a(7) = 132.

Then f(k) = 135*6^(k-2) - 3*2^(k-2) for k >= 0.

Proof: the formula is true for k = 0. Looking at the last 2 bits of n, it is easy to see that a(4n) = 2*a(2n), a(4n+1) = 4*a(2n)+3, a(4n+2) = 4*a(2n+1)+2 and a(4n+3) = 2*a(2n+1)+1. By summing over the recurrence relations for a(n), we get f(k+2) = Sum_{i=2^k..2^(k+1)-1} (f(4i) + f(4i+1) + f(4i+2) + f(4i+3)) = Sum_{i=2^k..2^(k+1)-1} (6a(2i) + 6a(2i+1) + 6) = 6*f(k+1) + 3*2^(k+1). Solving this first-order recurrence relation with the initial condition f(1) = 21 shows that f(k) = 135*6^(k-2) - 3*2^(k-2) for k > 0.

(End)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.

Formula

a(4n) = 2*a(2n), a(4n+1) = 4*a(2n)+3, a(4n+2) = 4*a(2n+1)+2 and a(4n+3) = 2*a(2n+1)+1. - Chai Wah Wu, Nov 25 2018

Example

6 in binary is 110. Modify each run by prepending a 1 to get 11110, which is 30 in decimal. So a(6) = 30.

Mathematica

Table[FromDigits[Flatten[Join[{1}, #]&/@Split[IntegerDigits[n, 2]]], 2], {n, 60}] (* Harvey P. Dale, Apr 26 2019 *)

Prog

(Python)
from re import split
def A320261(n):
    return int(''.join('1'+d for d in split('(0+)|(1+)', bin(n)[2:]) if d != '' and d != None), 2)

Crossrefs

Cf. A175046, A319423, A320037, A320038, A320039.

Sequence in context: A197423 A090786 A127818 * A344116 A187772 A291796

Adjacent sequences: A320258 A320259 A320260 * A320262 A320263 A320264

Keyword

nonn,base

Author

Chai Wah Wu, Oct 08 2018

Status

approved