

A320039


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


4



3, 13, 7, 25, 55, 29, 15, 49, 103, 221, 111, 57, 119, 61, 31, 97, 199, 413, 207, 441, 887, 445, 223, 113, 231, 477, 239, 121, 247, 125, 63, 193, 391, 797, 399, 825, 1655, 829, 415, 881, 1767, 3549, 1775, 889, 1783, 893, 447, 225, 455, 925, 463, 953, 1911, 957
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OFFSET

1,1


COMMENTS

A variation of A175046. Indices of record values are given by A319423.
From Chai Wah Wu, Nov 21 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) = 20.
Then f(k) = 21*6^(k1)  2^(k1) for k >= 0.
Proof: the equation for f is true for k = 0. Looking at the last 2 bits of n, it is easy to see that a(4n) = 2*a(2n)1, a(4n+1) = 4*a(2n)+3, a(4n+2) = 4*a(2n+1)+1 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) + 4) = 6*f(k+1) + 2^(k+2). Solving this firstorder recurrence relation with the initial condition f(1) = 20 shows that f(k) = 21*6^(k1)  2^(k1) 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)1, a(4n+1) = 4*a(2n)+3, a(4n+2) = 4*a(2n+1)+1 and a(4n+3) = 2*a(2n+1)+1.  Chai Wah Wu, Nov 21 2018


EXAMPLE

6 in binary is 110. Modify each run by appending a 1 to get 11101, which is 29 in decimal. So a(6) = 29.


MATHEMATICA

Array[FromDigits[Flatten@ Map[Append[#, 1] &, Split@ IntegerDigits[#, 2]], 2] &, 54] (* Michael De Vlieger, Nov 23 2018 *)


PROG

(Python)
from re import split
def A320039(n):
return int(''.join(d+'1' for d in split('(0+)(1+)', bin(n)[2:]) if d != '' and d != None), 2)


CROSSREFS

Cf. A175046, A319423, A320037, A320038.
Sequence in context: A230362 A173203 A140445 * A242173 A186109 A012789
Adjacent sequences: A320036 A320037 A320038 * A320040 A320041 A320042


KEYWORD

nonn,base


AUTHOR

Chai Wah Wu, Oct 05 2018


STATUS

approved



