A088431 -id:A088431 - cp's OEIS frontend

A007400 Continued fraction for Sum_{n>=0} 1/2^(2^n) = 0.8164215090218931...

0, 1, 4, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 6, 4

Offset: 0

Simon Plouffe, Jeffrey Shallit

			0.816421509021893143708079737... = 0 + 1/(1 + 1/(4 + 1/(2 + 1/(4 + ...))))

M. Kmošek, Rozwinieçie Niektórych Liczb Niewymiernych na Ulamki Lancuchowe (Continued Fraction Expansion of Some Irrational Numbers), Master's thesis, Uniwersytet Warszawski, 1979.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Henry Cohn, Symmetry and specializability in continued fractions, Acta Arithmetica, volume 75, number 4, 1996, pages 297-320 (PDF). Also arXiv:math/0008221 [math.NT].
W. F. Lunnon, Q-D Tables and Zero-Squares, Manuscript, Jan. 1974. (Annotated scanned copy)
R. M. MacGregor, Generalizing the notion of a periodic sequence, American Math. Monthly 87 (1980), 90-102. (Annotated scanned copy)
B. Salvy, Email to N. J. A. Sloane, May 1994
Ratpoly info, May 1994
Jeffrey Shallit, Simple continued fractions for some irrational numbers, J. Number Theory 11 (1979), no. 2, 209-217.
Jeffrey O. Shallit, Simple continued fractions for some irrational numbers, J. Number Theory 11 (1979), no. 2, 209-217.
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A007404 (decimal), A073088 (partial sums), A073414/A073415 (convergents), A088431 (half), A089267, A092910.

a:= proc(n) option remember; local n8, n16;
    n8:= n mod 8;
    if n8 = 0 or n8 = 3 then return 2
    elif n8 = 4 or n8 = 7 then return 4
    elif n8 = 1 then return procname((n+1)/2)
    elif n8 = 2 then return procname((n+2)/2)
    fi;
    n16:= n mod 16;
    if n16 = 5 or n16 = 14 then return 4
    elif n16 = 6 or n16 = 13 then return 6
    fi
end proc:
a(0):= 0: a(1):= 1: a(2):= 4:
map(a, [$0..1000]); # Robert Israel, Jun 14 2016

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]]]; Table[a[n], {n, 0, 98}] (* Jean-François Alcover, Nov 29 2013, after Ralf Stephan *)

a(n)=if(n<3,[0,1,4][n+1],if(n%8==1,a((n+1)/2),if(n%8==2,a((n+2)/2),[2,0,0,2,4,4,6,4,2,0,0,2,4,6,4,4][(n%16)+1]))) /* Ralf Stephan */

a(n)=contfrac(suminf(n=0,1/2^(2^n)))[n+1]

{ allocatemem(932245000); default(realprecision, 26000); x=suminf(n=0, 1/2^(2^n)); x=contfrac(x); for (n=1, 20001, write("b007400.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 07 2009

(define (A007400 n) (cond ((<= n 1) n) ((= 2 n) 4) (else (case (modulo n 8) ((0 3) 2) ((4 7) 4) ((1) (A007400 (/ (+ 1 n) 2))) ((2) (A007400 (/ (+ 2 n) 2))) (else (case (modulo n 16) ((5 14) 4) ((6 13) 6))))))) ;; (After Ralf Stephan's recurrence) - Antti Karttunen, Aug 12 2017

From Ralf Stephan, May 17 2005: (Start)
a(0)=0, a(1)=1, a(2)=4; for n > 2:
a(8k)    = a(8k+3)   = 2;
a(8k+4)  = a(8k+7)   = a(16k+5) = a(16k+14) = 4;
a(16k+6) = a(16k+13) = 6;
a(8k+1)  = a(4k+1);
a(8k+2)  = a(4k+2). (End)

A014707 a(4n) = 0, a(4n+2) = 1, a(2n+1) = a(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0

Offset: 0

N. J. A. Sloane

The regular paper-folding (or dragon curve) sequence.
It appears that the sequence of run lengths is A088431. - Dimitri Hendriks, May 06 2010
Runs of three consecutive ones appear around positions n = 22, 46, 54, 86, 94, 118, 150, 174, 182, ..., or for n of the form 2^(k+3)*(4*t+3)-2, k >= 0, t >= 0. - Vladimir Shevelev, Mar 19 2011

Guy Melançon, Factorizing infinite words using Maple, MapleTech journal, Vol. 4, No. 1, 1997, pp. 34-42, esp. p. 36.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Anna Lubiw, Michel France, Alfred van der Poorten, and Jeffrey Shallit, Convergents of folded continued fractions, Acta Arithmetica 77 (1996), 77-96.
Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jörg Endrullis, Dimitri Hendriks, and Jan Willem Klop, Degrees of streams, see table 1 "PF".
Jui-Yi Kao, Narad Rampersad, Jeffrey Shallit, and Manuel Silva, Words avoiding repetitions in arithmetic progressions, Theoretical Computer Science, Vol. 391, No. 1-2 (2008), pp. 126-137; arXiv preprint, arXiv:math/0608607 [math.CO], 2006.
Guy Melançon, Lyndon factorization of infinite words, STACS 96 (Grenoble, 1996), 147-154, Lecture Notes in Comput. Sci., 1046, Springer, Berlin, 1996. Math. Rev. 98h:68188.
Jeffrey Shallit, Cloitre's Self-Generating Sequence, arXiv:2501.00784 [math.CO], 2025.
László Tóth, Evaluating zeta(s) At Odd Positive Integers Using Automatic Dirichlet Series, arXiv:2508.04151 [math.NT], 2025.
Index entries for sequences obtained by enumerating foldings.

Equals 1 - A014577, which see for further references. Also a(n) = A038189(n+1).
The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410.
Cf. A220466

a014707 n = a014707_list !! n
a014707_list = f 0 $ cycle [0,0,1,0] where
   f i (x:_:xs) = x : a014707 i : f (i+1) xs
-- Reinhard Zumkeller, Sep 28 2011

nmax:=92: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := n mod 2 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 28 2013
# second Maple program:
a:= proc(n) option remember;
     `if`(n::even, irem(n/2, 2), a((n-1)/2))
    end:
seq(a(n), n=0..92);  # Alois P. Heinz, Jun 27 2022

a[n_ /; Mod[n, 4] == 0] = 0; a[n_ /; Mod[n, 4] == 2] = 1; a[n_ /; Mod[n, 2] == 1] := a[n] = a[(n - 1)/2]; Table[a[n],{n,0,92}] (* Jean-François Alcover, May 17 2011 *)
(1 - JacobiSymbol[-1, Range[100]])/2 (* Paolo Xausa, May 26 2024 *)

a(n)=n+=1;my(h=bitand(n,-n));n=bitand(n,h<<1);n!=0; \\ Joerg Arndt, Apr 09 2021

def A014707(n):
    s = bin(n+1)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return int(s[m-i-2]) if m-i-2 >= 0 else 0 # Chai Wah Wu, Apr 08 2021

def A014707(n): n+=1; h=n&-n; n=n&(h<<1); return int(n!=0)
print([A014707(n) for n in range(93)]) # Michael S. Branicky, Mar 29 2024 after Joerg Arndt

a(A091072(n)-1) = 0; a(A091067(n)-1) = 1. - Reinhard Zumkeller, Sep 28 2011 [Adjusted to match offset by Peter Munn, Jul 01 2022]
a(n) = (1-Jacobi(-1,n+1))/2 (cf. A034947). - N. J. A. Sloane, Jul 27 2012 [Adjusted to match offset by Peter Munn, Jul 01 2022]
Set a=0, b=1, S(0)=a, S(n+1) = S(n)aF(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).
a((2*n+1)*2^p-1) = n mod 2, p >= 0. - Johannes W. Meijer, Jan 28 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Aug 31 2024

More terms from Scott C. Lindhurst (ScottL(AT)alumni.princeton.edu)

A092910 a(n) is the (3n+2)-th component of the continued fraction for sum(k>=0,2^(-k!)).

Original entry on oeis.org

3, 4, 3, 3, 2, 3, 4, 3, 2, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 3, 2, 4, 3, 2, 3, 3, 4, 2, 3, 4, 3, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 3, 2, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 3, 4, 2, 3, 4, 3, 3, 2, 3, 4, 3, 2

Offset: 0

Benoit Cloitre, Apr 16 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A007400, A076157, A088431.

a(n)=5-component(contfrac(sum(i=0,10,1/2^(2^i))),n+3)/2

(define (A092910 n) (- 5 (* 1/2 (A007400 (+ 2 n))))) ;;  Code for A007400 given under that entry. - Antti Karttunen, Aug 12 2017

a(n) = 5 - (A007400(n+2)/2).

A088435 1/2 + half of the (n+1)-st component of the continued fraction expansion of sum(k>=1,1/3^(2^k)).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Nov 08 2003

nonn

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

			Example to illustrate the comment : a(a(1)+1) = a(4) = 2+(-1)^1 = 1 and a(2), a(3) are undefined. The rule forces a(2) = a(3) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..8192 (computed from the b-file of A004200 provided by _Harry J. Smith_)

Cf. A088431.

a(n) = (1/2) * (1+A004200(n+1)).

a(a(1)+a(2)+...+a(n)+1) = 2+(-1)^n.

A381929 Ending positions of runs in the regular paperfolding sequence A034947.

Original entry on oeis.org

2, 3, 5, 7, 10, 12, 13, 15, 18, 19, 21, 24, 26, 28, 29, 31, 34, 35, 37, 39, 42, 44, 45, 48, 50, 51, 53, 56, 58, 60, 61, 63, 66, 67, 69, 71, 74, 76, 77, 79, 82, 83, 85, 88, 90, 92, 93, 96, 98, 99, 101, 103, 106, 108, 109, 112, 114, 115, 117, 120, 122, 124, 125

Offset: 1

Jeffrey Shallit, Mar 10 2025

nonn

A "run" is a maximal block of consecutive identical terms.

			The first few terms of A034947 are 1,1,-1,1,1,-1,-1,1,1,1,-1, and the runs end at positions 2,3,5,7,10,... .

Martin Bunder, Bruce Bates, and Stephen Arnold, The summed paperfolding sequence, Bull. Aust. Math. Soc. 110 (2024), 189-198.
Jeffrey Shallit, Runs in Paperfolding Sequences, arXiv:2412.17930 [math.CO], 2024.

Cf. A034947. A371594 gives the starting positions of the runs, and A088431 gives the length of the runs.

cp's OEIS Frontend

A007400 Continued fraction for Sum_{n>=0} 1/2^(2^n) = 0.8164215090218931...

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A014707 a(4n) = 0, a(4n+2) = 1, a(2n+1) = a(n).

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A092910 a(n) is the (3n+2)-th component of the continued fraction for sum(k>=0,2^(-k!)).

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A088435 1/2 + half of the (n+1)-st component of the continued fraction expansion of sum(k>=1,1/3^(2^k)).

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A381929 Ending positions of runs in the regular paperfolding sequence A034947.

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