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A320262 Write n in binary, then modify each run of 0's and each run of 1's by appending a 0. a(n) is the decimal equivalent of the result. 2
2, 8, 6, 16, 34, 24, 14, 32, 66, 136, 70, 48, 98, 56, 30, 64, 130, 264, 134, 272, 546, 280, 142, 96, 194, 392, 198, 112, 226, 120, 62, 128, 258, 520, 262, 528, 1058, 536, 270, 544, 1090, 2184, 1094, 560, 1122, 568, 286, 192, 386, 776, 390, 784, 1570, 792, 398 (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) = 14.

Then f(k) = 15*6^(k-1) - 2^(k-1) 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), a(4n+1) = 4*a(2n)+2, a(4n+2) = 4*a(2n+1) and a(4n+3) = 2*a(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) = 14 shows that f(k) = 15*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(n) = 2*A320263(n).

a(4n) = 2*a(2n), a(4n+1) = 4*a(2n)+2, a(4n+2) = 4*a(2n+1) and a(4n+3) = 2*a(2n+1)+2. - Chai Wah Wu, Nov 21 2018

EXAMPLE

6 in binary is 110. Modify each run by appending a 0 to get 11000, which is 24 in decimal. So a(6) = 24.

MATHEMATICA

Array[FromDigits[Flatten@ Map[Append[#, 0] &, Split@ IntegerDigits[#, 2]], 2] &, 55] (* Michael De Vlieger, Nov 23 2018 *)

PROG

(Python)

from re import split

def A320262(n):

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

CROSSREFS

Cf. A175046, A319423, A320037, A320038, A320039, A320261.

Sequence in context: A197589 A124356 A277249 * A019186 A019187 A019243

Adjacent sequences:  A320259 A320260 A320261 * A320263 A320264 A320265

KEYWORD

nonn,base

AUTHOR

Chai Wah Wu, Oct 08 2018

STATUS

approved

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Last modified June 17 07:00 EDT 2019. Contains 324183 sequences. (Running on oeis4.)