A166242 - OEIS

%I #19 Feb 21 2021 13:56:08

%S 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,

%T 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,

%U 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

%N Sequence generated from A014577, the dragon curve.

%C 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.).

%H Antti Karttunen, <a href="/A166242/b166242.txt">Table of n, a(n) for n = -1..8191</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F 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).

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

%F Apparently, a(n) = 2^A000120(A003188(n+1)). - _Rémy Sigrist_, Feb 21 2021

%e From the Dragon curve, A014577:

%e ...1...1...0...1...1...0...0...1... generates A166242:

%e 1..2...4...2...4...8...4...2...4... given A166242(-1)=1.

%e 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,...).

%t 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 *)

%o (Scheme, with memoization-macro definec)

%o ;; Because definec does not work well with offset -1, we define an offset-0 based version of recurrence:

%o (definec (A166242off0 n) (if (zero? n) 1 (* (expt 2 (- (* 2 (A014577 (- n 1))) 1)) (A166242off0 (- n 1)))))

%o ;; which the offset -1 version will invoke:

%o (define (A166242 n) (A166242off0 (+ 1 n)))

%o (define (A014577 n) (- 1 (A000035 (/ (- (A000265 (+ 1 n)) 1) 2))))

%o ;; Scheme-code for A000035 and A000265 given under respective entries.

%o ;; _Antti Karttunen_, Jul 27 2017

%Y Cf. A000120, A003188, A014577, A164281.

%K nonn

%O -1,2

%A _Gary W. Adamson_, Oct 10 2009

%E More terms from _Antti Karttunen_, Jul 27 2017