A320916 -id:A320916 - cp's OEIS frontend

A019300 First n elements of Thue-Morse sequence A010060 read as a binary number.

0, 1, 3, 6, 13, 26, 52, 105, 211, 422, 844, 1689, 3378, 6757, 13515, 27030, 54061, 108122, 216244, 432489, 864978, 1729957, 3459915, 6919830, 13839660, 27679321, 55358643, 110717286, 221434573, 442869146, 885738292, 1771476585, 3542953171

Offset: 0

Jonas Wallgren

Ariel S Koiman, Table of n, a(n) for n = 0..3322

Cf. A010060, A048707, A320916 (bit reversal).

With[{tm=Nest[Flatten[#/.{0->{0,1},1->{1,0}}]&,{0},7]},Table[ FromDigits[ Take[tm,n],2],{n,40}]] (* Harvey P. Dale, Mar 25 2015 *)

a(n)=sum(k=1,n,(hammingweight(k)%2)<<(n-k)) \\ Charles R Greathouse IV, May 08 2016

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n,v[k]=2*v[k-1]+hammingweight(k)%2); concat(0,v) \\ Charles R Greathouse IV, May 08 2016

(define rdc(lambda(x)(if(null? (cdr x))'()(cons (car x) (rdc (cdr x))))))
; if a bit is 1, get 2^i, where i is the index of that bit from right-left
(define F (lambda (c i)(if (eq? c #\1) (expt 2 i) 0)))
; gathers the sum of 2^index for all indices corresponding to a 1
(define fn (lambda (x sum i stop)(if (eq? i stop) sum (fn (list->string (rdc (string->list x))) (+ sum (F (string-ref x (-(string-length x) 1)) i)) (+ i 1)stop))))
(define f (lambda (x)(fn (substring thue 0 (+ x 1)) 0 0 (string-length (substring thue 0 (+ x 1))) )))
(define thue "0110100110010110") ;Feel free to add Thue-Morse sequence of whatever length here
; Ariel S Koiman, May 07 2013

a(0) = 0, a(n+1) = 2a(n) + A010060(n). - Ralf Stephan, Sep 16 2003

A377029 a(1) = 0; thereafter in the binary expansion of a(n-1), expand bits: 1->01 and 0->10.

Original entry on oeis.org

0, 2, 6, 22, 406, 92566, 6818458006, 26055178074437806486, 540213899028732737068658940860686756246, 163551003506862550406254063077517364557434408527734307437037618419534882498966

Offset: 1

Darío Clavijo, Oct 13 2024

All terms are even and leading zeros omitted in the final encoding.
Conversely the opposite mapping of bits: 0->01 and 1->10 is A133468.
The bit length of a(n) is 2^(n-1)+1.
The count of bits set for a(n) is A094373(n).
a(n) = 2 (mod 4) for n > 1.
Also all the terms align bitwise to the right.
The hamming distance of a(n) and a(n+1) is in A000079.

			For n = 5 a(5) = 406 because:
This encoding results in the following tree:
n | a(n)
--+---------------
1 | 0
  | |\
2 | 1 0
  | | |
3 | 1 10
  | | | \
4 | 1 01 10--
  | | |\  \  \
  | | | \  \  \
5 | 1 10 01 01 10
Which also aligns bitwise to the right:
n | a(n)
--+-----------
1 |         0
2 |        10
3 |       110
4 |     10110
5 | 110010110
And 110010110 in base 10 is 406.

Darío Clavijo, Table of n, a(n) for n = 1..13

Cf. A000051, A000079, A000120, A094373, A133468, A320916, A374625.

NestList[FromDigits[2 - IntegerDigits[#, 2], 4] &, 0, 10] (* Paolo Xausa, Nov 04 2024 *)

from functools import cache
A374625 = lambda n: int(bin(n)[2:].replace('0', '2'), 4)
@cache
def a(n):
  if n == 1: return 0
  return A374625(a(n-1))
print([a(n) for n in range(1, 12)])

a(n) = A320916(2^(n-2)+1) for n > 1.
A000120(a(n+1) XOR a(n)) = A000079(n-2).
a(n) = A374625(a(n-1)) for n > 1. - Paolo Xausa, Nov 04 2024

cp's OEIS Frontend

A019300 First n elements of Thue-Morse sequence A010060 read as a binary number.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Scheme

Formula