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A320037 Write n in binary, then modify each run of 0's by appending one 1, and modify each run of 1's by appending one 0. a(n) is the decimal equivalent of the result. 6
2, 9, 6, 17, 38, 25, 14, 33, 70, 153, 78, 49, 102, 57, 30, 65, 134, 281, 142, 305, 614, 313, 158, 97, 198, 409, 206, 113, 230, 121, 62, 129, 262, 537, 270, 561, 1126, 569, 286, 609, 1222, 2457, 1230, 625, 1254, 633, 318, 193, 390, 793, 398, 817, 1638, 825, 414 (list; graph; refs; listen; history; text; internal format)
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) = 2 and f(1) = a(2) + a(3) = 15.

Then f(k) = 16*6^(k-1) - 2^(k-1) for k > 0.

Proof: looking at the last 2 bits of n, it is easy to see that a(4n) = 2a(2n)-1, a(4n+1) = 4a(2n)+2, a(4n+2) = 4a(2n+1)+1 and a(4n+3) = 2a(2n+1)+2. 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 first order recurrence relation with the initial condition f(1) = 15 shows that f(k) = 16*6^(k-1) - 2^(k-1) 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) = 2a(2n)-1, a(4n+1) = 4a(2n)+2, a(4n+2) = 4a(2n+1)+1 and a(4n+3) = 2a(2n+1)+2. - Chai Wah Wu, Nov 21 2018

EXAMPLE

6 in binary is 110. Modify each run by appending the opposite digit to get 11001, which is 25 in decimal. So a(6) = 25.

PROG

(Python)

from re import split

def A320037(n):

    return int(''.join(d+'0' if '1' in d else d+'1' for d in split('(0+)|(1+)', bin(n)[2:]) if d != '' and d != None), 2)

CROSSREFS

Cf. A175046, A319423, A320038.

Sequence in context: A011247 A201736 A068632 * A122664 A318649 A009306

Adjacent sequences:  A320034 A320035 A320036 * A320038 A320039 A320040

KEYWORD

nonn,base

AUTHOR

Chai Wah Wu, Oct 04 2018

STATUS

approved

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Last modified May 9 10:14 EDT 2021. Contains 343730 sequences. (Running on oeis4.)