A006697 -id:A006697 - cp's OEIS frontend

A134457 a(n) = number of length-n binary strings with A006697(n) distinct substrings.

1, 2, 2, 6, 8, 4, 18, 38, 48, 40, 16, 80, 210, 402, 644, 852, 928, 912, 704, 256, 1344, 3944, 9276, 19448, 37090, 65602, 107388, 160760, 220200

Offset: 0

David W. Wilson, Dec 17 2007

Here by "substring" one means a contiguous block within a string (often called "subword" or "factor").

			Table giving n, A006697(n) = maximal number of distinct substrings of a binary string of length n, a(n), c(n) = lexically first length-n binary string with A006697(n) distinct substrings.
n A006697(n) a(n) c(n)
0 1 1 null
1 2 2 0
2 4 2 01
3 6 6 001
4 9 8 0010
5 13 4 00110
6 17 18 000110
7 22 38 0001011
8 28 48 00010110
9 35 40 000101100
10 43 16 0001011100
11 51 80 00001011100
12 60 210 000010011101
13 70 402 0000100110111
14 81 644 00001001101110
15 93 852 000010011010111
16 106 928 0000100110101110
17 120 912 00001001101011100
18 135 704 000010011010111000
19 151 256 0000100110101111000
20 167 1344 00000100110101111000
21 184 3944 000001000110101111001
22 202 9276 0000010001100101111010
23 221 19448 00000100011001010111101
24 241 37090 000001000110010101111010
25 262 65602 0000010001100101001111011
26 284 107388 00000100011001010011101111
27 307 160760 000001000110010100111011110
28 331 220200 0000010001100101001110101111
For (conjectural?) further values see the _Martin Fuller_ link.

Martin Fuller, Algorithm and table of n, c(n) for n = 1..69.
Abraham Flaxman, Aram W. Harrow, Gregory B. Sorkin, Strings with Maximally Many Distinct Subsequences and Substrings, Electronic J. Combinatorics 11 (1) (2004), Paper R8.
Abraham Flaxman, Aram W. Harrow, Gregory B. Sorkin, Comments on the previous paper.
Antal Iványi, On the d-complexity of words, Ann. Univ. Sci. Budapest. Sect. Comput. 8 (1987), 69-90 (1988).
Jeffrey Shallit, On the maximum number of distinct factors in a binary string, Graphs Combin. 9 (1993), no. 2, 197-200.

Cf. A006697, A134466(n) = decimal value of c(n) interpreted as a binary number.

Link to comments on history of the problem added by Jeffrey Shallit, May 08 2016

A079559 Number of partitions of n into distinct parts of the form 2^j-1, j=1,2,....

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Jan 25 2003

Differences of the Meta-Fibonacci sequence for s=0. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)
Fixed point of morphism 0-->0, 1-->110 - Joerg Arndt, Jun 07 2007
A006697(k) gives number of distinct subwords of length k, conjectured to be equal to A094913(k)+1. - M. F. Hasler, Dec 19 2007
Characteristic function for the range of A005187: a(A055938(n))=0; a(A005187(n))=1; if a(m)=1 then either a(m-1)=1 or a(m+1)=1. - Reinhard Zumkeller, Mar 18 2009
The number of zeros between successive pairs of ones in this sequence is A007814. - Franklin T. Adams-Watters, Oct 05 2011
Length of n-th run = abs(A088705) + 1. - Reinhard Zumkeller, Dec 11 2011

			a(11)=1 because we have [7,3,1].
G.f. = 1 + x + x^3 + x^4 + x^7 + x^8 + x^10 + x^11 + x^15 + x^16 + x^18 + ...
From _Omar E. Pol_, Nov 30 2009: (Start)
The sequence, displayed as irregular triangle, in which rows length are powers of 2, begins:
1;
1,0;
1,1,0,0;
1,1,0,1,1,0,0,0;
1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0;
1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0;
1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0;
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Gary W. Adamson, Comments on A079559
Joerg Arndt, Matters Computational (The Fxtbook), section 1.26.5, Recursive generation and relation to a power series, page 74, figure 1.26-E and function A.
Benoit Cloitre, Fractal walk generated by the 130000 first terms (step of unit length moving to right if 0 left if 1) starting at (0,0)
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), R26.[alt source]
Thomas M. Lewis and Fabian Salinas, Optimal pebbling of complete binary trees and a meta-Fibonacci sequence, arXiv:2109.07328 [math.CO], 2021.
F. Ruskey and C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]
Index entries for characteristic functions

Cf. A000929, A001511, A005187, A006697, A007814, A046699, A055938, A094913, A163988.

Cf. A035263, A066519.

a079559 = p $ tail a000225_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Dec 11 2011

a079559_list = 1 : f [1] where
   f xs = ys ++ f ys where ys = init xs ++ [1] ++ tail xs ++ [0]
-- Reinhard Zumkeller, May 05 2015

g:=product(1+x^(2^n-1),n=1..15): gser:=series(g,x=0,110): seq(coeff(gser,x,n),n=0..104); # Emeric Deutsch, Apr 06 2006
d := n -> if n=1 then 1 else A046699(n)-A046699(n-1) fi; # Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

row[1] = {1}; row[2] = {1, 0}; row[n_] := row[n] = row[n-1] /. 1 -> Sequence[1, 1, 0]; Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Jul 30 2012, after Omar E. Pol *)
CoefficientList[ Series[ Product[1 + x^(2^n - 1), {n, 6}], {x, 0, 104}], x] (* or *)
Nest[ Flatten[# /. {0 -> {0}, 1 -> {1, 1, 0}}] &, {1}, 6] (* Robert G. Wilson v, Sep 08 2014 *)

w="1,";for(i=1,5,print1(w=concat([w,w,"0,"])))

A079559(n,w=[1])=until(n<#w=concat([w,w,[0]]),);w[n+1] \\ M. F. Hasler, Dec 19 2007

{a(n) = if( n<0, 0, polcoeff( prod(k=1, #binary(n+1), 1 + x^(2^k-1), 1 + x * O(x^n)), n))} /* Michael Somos, Aug 03 2009 */

def a053644(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1)
def a043545(n):
    x=bin(n)[2:]
    return int(max(x)) - int(min(x))
l=[1]
for n in range(1, 101): l+=[a043545(n + 1)*l[n + 1 - a053644(n + 1)], ]
print(l) # Indranil Ghosh, Jun 11 2017

G.f.: Product_{n>=1} (1 + x^(2^n-1)).
a(n) = 1 if n=0, otherwise A043545(n+1)*a(n+1-A053644(n+1)). - Reinhard Zumkeller, Aug 19 2006
a(n) = p(n,1) with p(n,k) = p(n-k,2*k+1) + p(n,2*k+1) if k <= n, otherwise 0^n. - Reinhard Zumkeller, Mar 18 2009
Euler transform is sequence A111113 sequence offset -1. - Michael Somos, Aug 03 2009
G.f.: Product_{k>0} (1 - x^k)^-A111113(k+1). - Michael Somos, Aug 03 2009
a(n) = A108918(n+1) mod 2. - Joerg Arndt, Apr 06 2011
a(n) = A000035(A153000(n)), n >= 1. - Omar E. Pol, Nov 29 2009, Aug 06 2013

Edited by M. F. Hasler, Jan 03 2008

A005942 a(2n) = a(n) + a(n+1), a(2n+1) = 2a(n+1), if n >= 2.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 20, 22, 24, 28, 32, 36, 40, 42, 44, 46, 48, 52, 56, 60, 64, 68, 72, 76, 80, 82, 84, 86, 88, 90, 92, 94, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

a(n) is the subword complexity (or factor complexity) of Thue-Morse sequence A010060, that is, the number of factors of length n in A010060. See Allouche-Shallit (2003). - N. J. A. Sloane, Jul 10 2012

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003. See Problem 10, p. 335. - From N. J. A. Sloane, Jul 10 2012
J. Berstel et al., Combinatorics on Words: Christoffel Words and Repetitions in Words, Amer. Math. Soc., 2008. See p. 83.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
A. de Luca and S. Varricchio, Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups, Theoret. Comput. Sci. 63 (1989), 333-348.
Jakub Konieczny and Clemens Müllner, Arithmetical subword complexity of automatic sequences, arXiv:2309.03180 [math.NT], 2023.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Luís M. S. Russo, Ana D. Correia, Gonzalo Navarro, and Alexandre P. Francisco, Approximating Optimal Bidirectional Macro Schemes, arXiv:2003.02336 [cs.DS], 2020.

Cf. A005943, A006697, A010060, A275202.

import Data.List (transpose)
a005942 n = a005942_list !! n
a005942_list = 1 : 2 : 4 : 6 : zipWith (+) (drop 6 ts) (drop 5 ts) where
   ts = concat $ transpose [a005942_list, a005942_list]
-- Reinhard Zumkeller, Nov 15 2012

a[0] = 1; a[1] = 2; a[2] = 4; a[3] = 6; a[n_?EvenQ] := a[n] = a[n/2] + a[n/2 + 1]; a[n_?OddQ]  := a[n] = 2*a[(n + 1)/2]; Array[a,60,0] (* Jean-François Alcover, Apr 11 2011 *)

a(n)=if(n<4,2*max(0,n)+(n==0),if(n%2,2*a((n+1)/2),a(n/2)+a(n/2+1)))

a(n) = 2*(A006165(n-1) + n - 1), n > 1.
G.f. (1+x^2)/(1-x)^2 + 2*x^2/(1-x)^2 * Sum_{k>=0} (x^2^(k+1)-x^(3*2^k)). - Ralf Stephan, Jun 04 2003
For n > 2, a(n) = 3*(n-1) + A053646(n-1). - Max Alekseyev, May 15 2011
a(n) = 2*A275202(n-1) for n > 1. - N. J. A. Sloane, Jun 04 2019

Typo in definition corrected by Reinhard Zumkeller, Nov 15 2012

A103354 a(n) = floor(x), where x is the solution to x = 2^(n-x).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 64, 65, 66, 67

Offset: 1

N. J. A. Sloane, Mar 21 2005

nonn

Partial sums seem to be A006697(n-1) = A094913(n-1) + 1. - M. F. Hasler, Dec 14 2007 [Confirmed by Allouche and Shallit, 2016. - N. J. A. Sloane, Mar 24 2017]

Robert Israel, Table of n, a(n) for n = 1..10000
J.-P. Allouche, J. Shallit, On the subword complexity of the fixed point of a -> aab, b -> b, and generalizations, arXiv preprint arXiv:1605.02361 [math.CO], 2016.

Cf. A006697 (partial sums), A094913.

A[1]:= 1;
for n from 2 to 100 do
  for x from A[n-1]  while x <= 2^(n-x) do od;
  A[n]:= x-1;
od:
seq(A[i],i=1..100); # Robert Israel, Dec 04 2016

a[n_] := Floor[ FullSimplify[ ProductLog[ 2^n*Log[2]]/Log[2]]]; Table[a[n], {n, 1, 74}] (* Jean-François Alcover, Dec 13 2011, after M. F. Hasler *)

A103354(n)=floor(solve(X=1,log(n)*2,X-2^(n-X))+1e-9) \\ M. F. Hasler, Dec 14 2007

A103354(n)=floor(n-log(n)/log(2)*(1-1.5/n)) \\ M. F. Hasler, Dec 14 2007

a(n) is approximately n/(1+log_2(n)/n).

a(n) = floor(LambertW(log(2)*2^n)/log(2)) = floor(n - log_2(n) + log_2(n)/(n log(2)) + O((log(n)/n)^2)) = floor(n - log_2(n) + 1.5*log_2(n)/n) at least for all n < 10^7. - M. F. Hasler, Dec 14 2007

Edited by M. F. Hasler, Dec 14 2007

A005943 Factor complexity (number of subwords of length n) of the Golay-Rudin-Shapiro binary word A020987.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 36, 46, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432, 440, 448, 456, 464, 472, 480, 488, 496, 504

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

Terms a(0)..a(13) were verified and terms a(14)..a(32) were computed using the first 2^32 terms of the GRS sequence. - Joerg Arndt, Jun 10 2012

Terms a(0)..a(63) were computed using the first 2^36 terms of the GRS sequence, and are consistent with Arndt's conjectured g.f. - Sean A. Irvine, Oct 12 2016

			All 8 subwords of length three (000, 001, ..., 111) occur in A020987, so a(3) = 8.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David A. Corneth, Table of n, a(n) for n = 0..9999
Jean-Paul Allouche, The Number of Factors in a Paperfolding Sequence, Bulletin of the Australian Mathematical Society, volume 46, number 1, August 1992, pages 23-32. Section 6 theorem 2, a(n) = P_{w_i}(n).
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A006697, A005942, A337120 (paperfolding).

# Naive Maple program, useful for getting initial terms of factor complexity FC of a sequence b1[]. N. J. A. Sloane, Jun 04 2019
FC:=[0]; # a(0)=0 from the empty subword
for L from 1 to 12 do
  lis := {};
  for n from 1 to nops(b1)-L do
    s:=[seq(b1[i],i=n..n+L-1)];
    lis:={op(lis),s}; od:
FC:=[op(FC),nops(lis)];
od:
FC;

CoefficientList[Series[(1 + x^2 + 2 x^3 + 4 x^4 + 4 x^6 - 2 x^7 - 2 x^9)/(1 - x)^2, {x, 0, 64}], x] (* Michael De Vlieger, Oct 14 2021 *)

first(n) = n = max(n, 10); concat([1, 2, 4, 8, 16, 24, 36, 46], vector(n-8,i,8*i+48)) \\ David A. Corneth, Apr 28 2021

G.f.: (1+x^2+2*x^3+4*x^4+4*x^6-2*x^7-2*x^9)/(1-x)^2. - Joerg Arndt, Jun 10 2012
From Kevin Ryde, Aug 18 2020: (Start)
a(1..7) = 2,4,8,16,24,36,46, then a(n) = 8*n - 8 for n>=8. [Allouche]
a(n) = 2*A337120(n-1) for n>=1. [Allouche, end of proof of theorem 2]
(End)

Minor edits by N. J. A. Sloane, Jun 06 2012
a(14)-a(32) added by Joerg Arndt, Jun 10 2012
a(33)-a(36) added by Joerg Arndt, Oct 28 2012

A094913 Maximal number of distinct nonempty substrings of any binary string of length n.

Original entry on oeis.org

0, 1, 3, 5, 8, 12, 16, 21, 27, 34, 42, 50, 59, 69, 80, 92, 105, 119, 134, 150, 166, 183, 201, 220, 240, 261, 283, 306, 330

Offset: 0

John W. Layman, Jun 17 2004

It would be more natural to include the empty substring, which would result in the sequence b(n)=a(n)+1; all values computed so far confirm the conjecture that a(n)+1 = A006697(n). - M. F. Hasler, Dec 17 2007
In addition to the example given, computation finds 17 other binary strings of length 6 which contain the maximal number a(6)=16 of distinct substrings. Interestingly, each of the 18 such instances gives the same numbers of substrings of each possible length, in this case 2,4,4,3,2,1 substrings of lengths 1 through 6, respectively.
Calculations suggest that, for any n>0, binary strings of length n exist such that the number of distinct substrings of length k, 1<=k<=n, is as large as possible consistent with basic counting principles, i.e., n-k+1 substrings of length k starting at each of the n-k+1 possible starting positions, subject to the condition, however, that there cannot be more than 2^k distinct binary strings of length k. This suggests the following conjecture. Conjecture: The maximum number a(n) of distinct substrings of a binary string of length n is given by a(n) = Sum_{k=1..n} t(k) where t(k) is the smaller of 2^k and n-k+1.
Conjecture: a(n) = A006697(n)-1. - Vladeta Jovovic, Sep 19 2005
Conjecture: a(n) = 2^(m+1)-2 + (n-m)(n-m+1)/2, where m = floor(n+1-LambertW( 2^(n+1) * log(2) ) / log(2) ) = integer part of the solution to 2^x = n+1-x. This is based on the reasoning that you can construct the word of length n so that it contains the maximal possible number, max( n-k+1, 2^k ), of different substrings of length k. - M. F. Hasler, Dec 17 2007
From Peter Pein (petsie(AT)dordos.net), Jan 18 2008: (Start)
The following heuristic seems to work for computing the maximal number of distinct substrings of a binary string of length n.
Start with the empty list and at each step try to insert a zero or a one at each possible position. Then pick the first list with the maximal number of sublists and start over.
Say we have had {0,0,1,1,0} as one of the lists with the maximal number of sublists. Then my candidates for the next test are:
With[{lastbest = {0, 0, 1, 1, 0}}, Union[Flatten[ Outer[Insert[lastbest, #2, #1] &, Range[1 + Length[lastbest]], {0, 1}, 1], 1]]]
{{0, 0, 0, 1, 1, 0}, {0, 0, 1, 0, 1, 0}, {0, 0, 1, 1, 0, 0}, {0, 0, 1, 1, 0, 1}, {0, 0, 1, 1, 1, 0}, {0, 1, 0, 1, 1, 0}, {1, 0, 0, 1, 1, 0}}
See http://freenet-homepage.de/Peter_Berlin/Mathe/heuristicA094913.nb for the Mathematica notebook with the complete (simple) algorithm. There's a screenshot too (same URL but with .png instead of .nb).
If this works correctly, the first 100 values of A094913 are calculated in 30 secs.
(End) [See also the remarks in the Fuller link in A134457.]
From Jon Perry, Nov 16 2010: (Start)
Conjecture: column sums of:
1 3 5 7 9  11 13 ...
      1 3   5  7 ...
               1 ...
....................
--------------------
1 3 5 8 12 16 21 ... (End)

			The binary string 000110 of length 6 contains the 16 distinct substrings 0, 1, 00, 01, 11, 10, 000, 001, 011, 110, 0001, 0011, 0110, 00011, 00110, 000110 and a computer search shows that no other binary string of length 6 contains more than 16. Thus a(6)=16.
G.f. = x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 16*x^6 + 21*x^7 + 27*x^8 + 34*x^9 + ...

The history of this problem is given in a comment in the Electronic Journal of Combinatorics. Some of the conjectures above are proven in the papers cited.
J.-P. Allouche, J. Shallit, On the subword complexity of the fixed point of a -> aab, b -> b, and generalizations, arXiv preprint arXiv:1605.02361 [math.CO], 2016.
Abraham Flaxman, Aram W. Harrow, Gregory B. Sorkin, Strings with Maximally Many Distinct Subsequences and Substrings, Electronic J. Combinatorics 11 (1) (2004), Paper R8.
Antal Iványi, On the d-complexity of words, Ann. Univ. Sci. Budapest. Sect. Comput. 8 (1987), 69-90 (1988).
Jeffrey Shallit, On the maximum number of distinct factors in a binary string, Graphs Combin. 9 (1993), no. 2, 197-200.

Cf. A006697, A134457, A134466.

A103354[n_] := Floor[ FullSimplify[ ProductLog[ 2^n*Log[2]]/Log[2]]]; Accumulate[ Table[ A103354[n], {n, 1, 29}]] - 1 (* Jean-François Alcover, Dec 13 2011, after M. F. Hasler *)

a(19)-a(28) from David W. Wilson, Dec 17 2007

More references (some proving some conjectures) added by Jeffrey Shallit, Nov 21 2015

cp's OEIS Frontend

A134457 a(n) = number of length-n binary strings with A006697(n) distinct substrings.

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A079559 Number of partitions of n into distinct parts of the form 2^j-1, j=1,2,....

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A005942 a(2n) = a(n) + a(n+1), a(2n+1) = 2a(n+1), if n >= 2.

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A103354 a(n) = floor(x), where x is the solution to x = 2^(n-x).

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A005943 Factor complexity (number of subwords of length n) of the Golay-Rudin-Shapiro binary word A020987.

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Programs

Maple

Mathematica

PARI

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A094913 Maximal number of distinct nonempty substrings of any binary string of length n.

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