A166242 Sequence generated from A014577, the dragon curve.

1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2, 4, 8, 16, 8, 16, 32, 16, 8, 4, 8, 16, 8, 4, 8, 4, 2, 4, 8, 16, 8, 16, 32, 16, 8, 16, 32, 64, 32, 16, 32, 16, 8, 4, 8, 16, 8, 16, 32, 16, 8, 4, 8, 16, 8, 4, 8, 4, 2, 4, 8, 16, 8, 16, 32, 16, 8, 16, 32, 64, 32, 16, 32, 16, 8, 16, 32, 64, 32, 64, 128, 64, 32, 16, 32

Offset

-1,2

Comments

Rows of A164281 tend to A166242. Subsets of the first 2^n terms can be parsed into a binomial frequency of powers of 2; for example, the first 16 terms has as frequency of (1, 4, 6, 4, 1): (one 1, four 2's, six 4's, four 8's, and one 16.).

Links

Antti Karttunen, Table of n, a(n) for n = -1..8191

Index entries for sequences related to binary expansion of n

Formula

Let a(-1) = 1, then a(n+1) = 2*a(n) if A014577(n+1) = 1. If A014577(n+1) = 0, then a(n+1) = (1/2)*a(n).

As a recursive string in subsets of 2^n terms, the next subset = twice each term of current string, reversed, and appended.

Apparently, a(n) = 2^A000120(A003188(n+1)). - Rémy Sigrist, Feb 21 2021

Example

From the Dragon curve, A014577:
...1...1...0...1...1...0...0...1... generates A166242:
1..2...4...2...4...8...4...2...4... given A166242(-1)=1.
By recursion, given the first four terms: (1, 2, 4, 2); reverse, double, and append to (1, 2, 4, 2) getting (1, 2, 4, 2, 4, 8, 4, 2,...).

Mathematica

FoldList[If[EvenQ[((#2 + 1)/2^IntegerExponent[#2 + 1, 2] - 1)/2], 2 #1, #1/2] &, 1, Range[0, 89]] (* Michael De Vlieger, Jul 29 2017, after Jean-François Alcover at A014577 *)

Prog

(Scheme, with memoization-macro definec)
;; Because definec does not work well with offset -1, we define an offset-0 based version of recurrence:
(definec (A166242off0 n) (if (zero? n) 1 (* (expt 2 (- (* 2 (A014577 (- n 1))) 1)) (A166242off0 (- n 1)))))
;; which the offset -1 version will invoke:
(define (A166242 n) (A166242off0 (+ 1 n)))
(define (A014577 n) (- 1 (A000035 (/ (- (A000265 (+ 1 n)) 1) 2))))
;; Scheme-code for A000035 and A000265 given under respective entries.
;; Antti Karttunen, Jul 27 2017

Crossrefs

Cf. A000120, A003188, A014577, A164281.

Sequence in context: A296141 A286536 A318768 * A143107 A051638 A286580

Adjacent sequences: A166239 A166240 A166241 * A166243 A166244 A166245

Keyword

nonn

Author

Gary W. Adamson, Oct 10 2009

Extensions

More terms from Antti Karttunen, Jul 27 2017

Status

approved