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A166242
Sequence generated from A014577, the dragon curve.
6
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.).
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
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Oct 10 2009
EXTENSIONS
More terms from Antti Karttunen, Jul 27 2017
STATUS
approved