

A166242


Sequence generated from A014577, the dragon curve.


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 JeanFrançois Alcover at A014577 *)


PROG

(Scheme, with memoizationmacro definec)
;; Because definec does not work well with offset 1, we define an offset0 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))))
;; Schemecode 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



