A088431 Half of the (n+1)-st component of the continued fraction expansion of Sum_{k>=0} 1/2^(2^k).

2, 1, 2, 2, 3, 2, 1, 2, 3, 1, 2, 3, 2, 2, 1, 2, 3, 1, 2, 2, 3, 2, 1, 3, 2, 1, 2, 3, 2, 2, 1, 2, 3, 1, 2, 2, 3, 2, 1, 2, 3, 1, 2, 3, 2, 2, 1, 3, 2, 1, 2, 2, 3, 2, 1, 3, 2, 1, 2, 3, 2, 2, 1, 2, 3, 1, 2, 2, 3, 2, 1, 2, 3, 1, 2, 3, 2, 2, 1, 2, 3, 1, 2, 2, 3, 2, 1, 3, 2, 1, 2, 3, 2, 2, 1, 3, 2, 1, 2, 2, 3, 2, 1, 2, 3

Offset

1,1

Comments

To construct the sequence use the rule: a(1)=2, then a(a(1) + a(2) + ... + a(n) + 1) = 2 and fill in any undefined places with the sequence 1,3,1,3,1,3,1,3,1,3,1,3,....

This sequence appears to be the sequence of run lengths of the regular paperfolding sequence A014577, i.e., the latter starts as follows: 2 zeros, 1 one, 2 zeros, 2 ones, etc. - Dimitri Hendriks, May 06 2010

Hendriks' conjecture is proved in Bunder, Bates, and Arnold (2024). Also see Shallit (2024). - Jeffrey Shallit, Mar 10 2025

Links

Antti Karttunen, Table of n, a(n) for n = 1..8192

Martin Bunder, Bruce Bates, and Stephen Arnold, The summed paperfolding sequence, Bull. Austral. Math. Soc. 110 (2024), 189-198.

Kevin Ryde, Iterations of the Dragon Curve, see index "TurnRun", with a(n) = TurnRun(n-1).

Jeffrey Shallit, Runs in Paperfolding Sequences, arXiv:2412.17930 [math.CO], 2024. See p. 10.

Formula

a(n) = (1/2)*A007400(n+1); a(a(1) + a(2) + ... + a(n) + 1) = 2.

Example

Example to illustrate the comment: a(a(1)+1)=a(3)=2 and a(2) is undefined. The rule requires a(2)=1. Next, a(a(1)+a(2)+1)=a(4)=2, a(a(1)+a(2)+a(3)+1)=a(6)=2 and a(5) is undefined. The rule now requires a(5)=3.

Mathematica

a[n_] := a[n] = Which[n < 3, {0, 1, 4}[[n + 1]], Mod[n, 8] == 1, a[(n + 1)/2], Mod[n, 8] == 2, a[(n + 2)/2], True, {2, 0, 0, 2, 4, 4, 6, 4, 2, 0, 0, 2, 4, 6, 4, 4}[[Mod[n, 16] + 1]]]; Array[a[# + 1]/2 &, 98] (* after Jean-François Alcover at A007400 *)

Prog

(Scheme) (define (A088431 n) (* 1/2 (A007400 (+ 1 n)))) ;; Code for A007400 given under that entry. - Antti Karttunen, Aug 12 2017

Crossrefs

Cf. A007400, A014577, A088435, A092910.

Sequence in context: A380117 A284044 A126305 * A254661 A052304 A049874

Adjacent sequences: A088428 A088429 A088430 * A088432 A088433 A088434

Keyword

nonn,changed

Author

Benoit Cloitre, Nov 08 2003

Status

approved