A137844 - OEIS

%I #20 Oct 21 2017 22:05:24

%S 1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,5,1,1,1,

%T 2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,6,1,1,1,2,1,1,

%U 1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,5,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1

%N Define S(1) = {1}, S(n+1) = S(n) U S(n) if a(n) is even, S(n+1) = S(n) U n U S(n) if a(n) is odd. Sequence {a(n), n >= 1} is limit as n approaches infinity of S(n). (U represents concatenation.).

%H Antti Karttunen, <a href="/A137844/b137844.txt">Table of n, a(n) for n = 1..31739, up to the stage S(15)</a>

%e S(1) = {1}.

%e S(2) = {1,1,1}, because a(1) = 1, which is odd.

%e S(3) = {1,1,1,2,1,1,1}, because a(2) = 1, which is odd.

%e S(4) = {1,1,1,2,1,1,1,3,1,1,1,2,1,1,1}.

%e S(5) = {1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1}, because a(4) = 2, which is even.

%e Etc.

%t Fold[Flatten@ Join[#1, If[OddQ[#1[[#2]]], {#2}, {}], #1] &, {1}, Range@ 6] (* _Michael De Vlieger_, Oct 18 2017 *)

%o (Scheme, with memoization-macro definec)

%o (definec (A137844 n) (if (= 1 n) n (let ((k (let loop ((j 1)) (if (>= (A291754 j) n) j (loop (+ 1 j)))))) (cond ((= (+ 1 (A291754 (- k 1))) n) (if (odd? (A137844 (- k 1))) (- k 1) 1)) (else (A137844 (- n (+ (A291754 (- k 1)) (A000035 (A137844 (- k 1)))))))))))

%o (definec (A291754 n) (if (= 1 n) 1 (+ (* 2 (A291754 (- n 1))) (A000035 (A137844 (- n 1))))))

%o (define (A000035 n) (modulo n 2))

%o ;; _Antti Karttunen_, Aug 31 2017

%Y Cf. A137843, A291754 (the length of stage n).

%K easy,nonn

%O 1,4

%A _Leroy Quet_, Feb 13 2008

%E Data section filled up to the length of stage S(7) by _Antti Karttunen_, Aug 31 2017