A133468 -id:A133468 - cp's OEIS frontend

A048707 Numerators of ratios converging to Thue-Morse constant.

0, 1, 6, 105, 27030, 1771476585, 7608434000728254870, 140350834813144189858090274002849666665, 47758914269546354982683078068829456704164423862093743397580034411621752859030

Offset: 0

Antti Karttunen, Mar 09 1999

Also interpret each iteration of the construction of the Thue-Morse constant as a binary number converted to a decimal number. Thus (0_b, 01_b, 0110_b, 01101001_b ...) gives the present sequence in decimal. - Robert G. Wilson v, Sep 22 2006

a(n) corresponds to the binary value of the truth-table for the xor operator with n-arguments. - Joe Riel (joer(AT)san.rr.com), Jan 31 2010

Ariel S Koiman, Table of n, a(n) for n = 0..11 (shortened by _N. J. A. Sloane_, Jan 13 2019)
Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 122 (Schroeppel, Gosper)
Eric Weisstein's World of Mathematics, Thue-Morse Sequence

The denominators are given by A001146. Consists of every 2^n-th term of A019300. Cf. A048708 (same sequence in hexadecimal) and A014571, A010060, A014572.

Cf. A080814, A080815, A133468.

Table[ FromDigits[ Nest[ Flatten[ #1 /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, n], 2], {n, 0, 8}] (* Robert G. Wilson v, Sep 22 2006 *)

;returns all but the last element of a list
(define rdc(lambda(x)(if(null? (cdr x))'()(cons (car x) (rdc (cdr x))))))
;gets the two's complement of a given bit
(define twosComplement (lambda (x)(if (eq? x #\0) "1" "0" )))
;gets the two's complement of a string
(define complementOfCurrent (lambda (x y z)(if (eq? (string-length y) z) y (complementOfCurrent (list->string (cdr (string->list x))) (string-append y (twosComplement (string-ref x 0))) z))))
;concatenates the two's complement of a string onto the current string, giving the next element in the TM sequence
(define concatenateComplement (lambda (x i)(if(zero? i) x (concatenateComplement(string-append x (complementOfCurrent x "" (string-length x)))(- i 1)))))
;generates the TM sequence of length 2^x
(define generateThue (lambda (x)(concatenateComplement "0" 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 (generateThue x) 0 0 (string-length (generateThue x)))))
;format: (f x)
;example: (f 10)
;by Ariel S Koiman, Apr 23 2013

a(0) = 0, a(n) = (a(n-1)+1)*((2^(2^(n-1)))-1).

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

A337672 Numbers with binary expansion Sum_{k = 0..w} b_k * 2^k such that the polynomial Sum_{k = 0..w} (X+k)^2 * (-1)^b_k is constant.

Original entry on oeis.org

0, 9, 150, 153, 165, 195, 2268, 2282, 2289, 2364, 2394, 2406, 2409, 2454, 2457, 2469, 2499, 2618, 2646, 2649, 2661, 2702, 2709, 2723, 2829, 2835, 3126, 3129, 3150, 3157, 3171, 3213, 3219, 3339, 3591, 34680, 34740, 34764, 34770, 34785, 35576, 35700, 35756

Offset: 1

Rémy Sigrist, Sep 15 2020

Leading 0's in binary expansions are ignored.
Positive terms are digitally balanced (A031443).
If m belongs to the sequence, then A056539(m) also belongs to the sequence.
If m and n belong to the sequence, then their binary concatenation also belongs to the sequence (assuming the concatenation with 0 is neutral).

			The first 16 integers, alongside their binary representations and associate polynomials, are:
  k   bin(k)  P(k)
  --  ------  --------------
   0       0               0
   1       1            -X^2
   2      10           2*X+1
   3      11    -2*X^2-2*X-1
   4     100       X^2+6*X+5
   5     101      -X^2-2*X-3
   6     110      -X^2+2*X+3
   7     111    -3*X^2-6*X-5
   8    1000   2*X^2+12*X+14
   9    1001              -4
  10    1010           4*X+6
  11    1011   -2*X^2-8*X-12
  12    1100          8*X+12
  13    1101    -2*X^2-4*X-6
  14    1110        -2*X^2+4
  15    1111  -4*X^2-12*X-14
We have constant polynomials for k = 0 and k = 9, so a(1) = 0 and a(2) = 9.

Index entries for sequences related to binary expansion of n

Cf. A031443, A056539, A133468.

is(n) = { my (b=Vecrev(binary(n))); poldegree(p=sum(k=1, #b, ('X+k-1)^2 * (-1)^b[k]))<=0 }

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A048707 Numerators of ratios converging to Thue-Morse constant.

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

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A337672 Numbers with binary expansion Sum_{k = 0..w} b_k * 2^k such that the polynomial Sum_{k = 0..w} (X+k)^2 * (-1)^b_k is constant.

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