A135416 -id:A135416 - cp's OEIS frontend

A091090 In binary representation: number of editing steps (delete, insert, or substitute) to transform n into n + 1.

1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2

Offset: 0

Reinhard Zumkeller, Dec 19 2003

Apparently, one less than the number of cyclotomic factors of the polynomial x^n - 1. - Ralf Stephan, Aug 27 2013

Let the binary expansion of n >= 1 end with m >= 0 1's. Then a(n) = m if n = 2^m-1 and a(n) = m+1 if n > 2^m-1. - Vladimir Shevelev, Aug 14 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 11.
Michael Gilleland, Levenshtein Distance [broken link] [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Ryota Inagaki, Tanya Khovanova, and Austin Luo, On Chip-Firing on Undirected Binary Trees, Ann. Comb. (2025). See p. 22.
Frank Ruskey and Chris Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009), Article 09.4.3.
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.
Eric Weisstein's World of Mathematics, Binary.
Eric Weisstein's World of Mathematics, Binary Carry Sequence.
WikiBooks: Algorithm Implementation, Levenshtein distance.
Index entries for sequences related to binary expansion of n.

Cf. A007088, A135517.

This is Guy Steele's sequence GS(2, 4) (see A135416).

a091090 n = a091090_list !! n
a091090_list = 1 : f [1,1] where f (x:y:xs) = y : f (x:xs ++ [x,x+y])
-- Same list generator function as for a036987_list, cf. A036987.
-- Reinhard Zumkeller, Mar 13 2011

a091090' n = levenshtein (show $ a007088 (n + 1)) (show $ a007088 n) where
  levenshtein :: (Eq t) => [t] -> [t] -> Int
  levenshtein us vs = last $ foldl transform [0..length us] vs where
    transform xs@(x:xs') c = scanl compute (x+1) (zip3 us xs xs') where
      compute z (c', x, y) = minimum [y+1, z+1, x + fromEnum (c' /= c)]
-- Reinhard Zumkeller, Jun 09 2015

-- following Vladeta Jovovic's formula
import Data.List (transpose)
a091090'' n = vjs !! n where
   vjs = 1 : 1 : concat (transpose [[1, 1 ..], map (+ 1) $ tail vjs])
-- Reinhard Zumkeller, Jun 09 2015

A091090 := proc(n)
    if n = 0 then
        1;
    else
        A007814(n+1)+1-A036987(n) ;
    end if;
end proc:
seq(A091090(n),n=0..100); # R. J. Mathar, Sep 07 2016
# Alternatively, explaining the connection with A135517:
a := proc(n) local count, k; count := 1; k := n;
while k <> 1 and k mod 2 <> 0 do count := count + 1; k := iquo(k, 2) od:
count end: seq(a(n), n=0..101); # Peter Luschny, Aug 10 2017

a[n_] := a[n] = Which[n==0, 1, n==1, 1, EvenQ[n], 1, True, a[(n-1)/2] + 1]; Array[a, 102, 0] (* Jean-François Alcover, Aug 12 2017 *)

a(n)=my(m=valuation(n+1,2)); if(n>>m, m+1, max(m, 1)) \\ Charles R Greathouse IV, Aug 15 2017

def A091090(n): return (~(n+1)&n).bit_length()+bool(n&(n+1)) if n else 1 # Chai Wah Wu, Sep 18 2024

a(n) = LevenshteinDistance(A007088(n), A007088(n + 1)).
a(n) = A007814(n + 1) + 1 - A036987(n).
a(n) = A152487(n + 1, n). - Reinhard Zumkeller, Dec 06 2008
a(A004275(n)) = 1. - Reinhard Zumkeller, Mar 13 2011
From Vladeta Jovovic, Aug 25 2004, fixed by Reinhard Zumkeller, Jun 09 2015: (Start)
a(2*n) = 1, a(2*n + 1) = a(n) + 1 for n > 0.
G.f.: 1 + Sum_{k > 0} x^(2^k - 1)/(1 - x^(2^(k - 1))). (End)
Let T(x) be the g.f., then T(x) - x*T(x^2) = x/(1 - x). - Joerg Arndt, May 11 2010
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Sep 29 2023
a(n) = A000120(n) + A070939(n) - A000120(n+1) - A070939(n+1) + 2. - Chai Wah Wu, Sep 18 2024

A008687 Number of 1's in 2's complement representation of -n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 7, 6, 6, 5, 6, 5, 5, 4, 6, 5, 5, 4, 5, 4, 4, 3, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3

Offset: 0

R. H. Hardin

a(A127904(n)) = n and a(m) < n for m < A127904(n). - Reinhard Zumkeller, Feb 05 2007
a(n) = A000120(A010078(n)), n>0; a(n) = A023416(A004754(n-1)), n>1. - Reinhard Zumkeller, Dec 04 2015
Conjecture: a(n)+1 is the length of the Hirzebruch (negative) continued fraction for the Stern-Brocot tree fraction A007305(n)/A007306(n). - Andrey Zabolotskiy, Apr 17 2020
Terms a(n); n >= 2 can be generated recursively, as follows. Let S(0) = {1}, then for k >=1, let S(k) = {S(k-1)+1, S(k-1)}, where +1 means +1 on every term of S(k-1); see Example. Each step of the recursion gives the next 2^k terms of the sequence. - David James Sycamore, Jul 15 2024

			Using the above recursion for a(n); n >= 2, we have:
 S(0) = {1} so a(2) = 1;
 S(1) = {2,1} so a(3,4) = 2,1;
 S(2) = {3,2,2,1}, so a(5,6,7,8) = 3,2,2,1;
As irregular table the sequence for n >= 2 begins:
  1;
  2,1;
  3,2,2,1;
  4,3,3,2,3,2,2,1;
  5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1;
  6,5,5,4,5,4,4,3,5,4,3,3,4,3,3,2,5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1;
and so on (the k-th row contains 2^k terms; k>=0). - _David James Sycamore_, Jul 15 2024

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Wikipedia, Two's complement

This is Guy Steele's sequence GS(4, 3) (see A135416).
Cf. A000120, A004754, A010078, A023416, A290251.
Cf. A007305, A007306, A061313, A088696.

a008687 n = a008687_list !! n
a008687_list = 0 : 1 : c [1] where c (e:es) = e : c (es ++ [e+1,e])
-- Reinhard Zumkeller, Mar 07 2011

a[0] = 0; a[1] = 1; a[n_] := a[n] = Mod[n,2] + a[Mod[n,2] + Floor[n/2]]; Array[a, 96, 0] (* Jean-François Alcover, Aug 12 2017, after Reinhard Zumkeller *)

a(n) = if(n<2,n, n--; logint(n,2) - hammingweight(n) + 2); \\ Kevin Ryde, Apr 14 2021

a(n) = A023416(n-1) + 1.
a(n) = if n<=1 then n else (n mod 2) + a((n mod 2) + floor(n/2)). - Reinhard Zumkeller, Feb 05 2007
a(n) = if n<2 then n else a(ceiling(n/2)) + n mod 2. - Reinhard Zumkeller, Jul 25 2006
Min{m: a(m)=n} = if n>0 then A083318(n-1) else 0. - Reinhard Zumkeller, Jul 25 2006

A135528 1, then repeat 1,0.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

This is Guy Steele's sequence GS(2, 1) (see A135416).
2-adic expansion of 1/3 (right to left): 1/3 = ...01010101010101011. - Philippe Deléham, Mar 24 2009
Also, with offset 0, parity of A036467(n-1). - Omar E. Pol, Mar 17 2015
Appears to be the Gilbreath transform of 1,2,3,5,7,11,13,... (A008578). (This is essentially the same as the Gilbreath conjecture, see A036262.) - N. J. A. Sloane, May 08 2023

			G.f. = x + x^2 + x^4 + x^6 + x^8 + x^10 + x^12 + x^14 + x^16 + x^18 + x^20 + ...

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A008578, A036467, A049711, A135416.

a135528 n = a135528_list !! (n-1)
a135528_list = concat $ iterate ([1,0] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (0:ps) * qs         = 0 : ps * qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

Maple
```
GS(2,1,200); [see A135416].
```

Prepend[Table[Mod[n + 1, 2], {n, 2, 60}], 1] (* Michael De Vlieger, Mar 17 2015 *)
PadRight[{1},120,{0,1}] (* Harvey P. Dale, Apr 23 2024 *)

G.f.: x*(1+x-x^2)/(1-x^2). - Philippe Deléham, Feb 08 2012
G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x)))). - Michael Somos, Apr 02 2012
a(n) = A049711(n+2) mod 2. - Ctibor O. Zizka, Jan 28 2019

A135481 a(n) = 2^A007814(n+1) - 1.

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 63, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1, 0

Offset: 0

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

This is Guy Steele's sequence GS(1, 6) (see A135416).

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A006519, A007814, A135416, A140670, A136013 (partial sums).

using IntegerSequences
A135481List(len) = [Bits("CNIMP", n+1, n) for n in 0:len]
println(A135481List(100))  # Peter Luschny, Sep 25 2021

Maple
```
GS(1,6,200); # see A135416
```

Table[BitAnd[i, BitNot[i+1]], {i, 0, 200}] (* Peter Luschny, Jun 01 2011 *)

a(n) = 2^valuation(n+1, 2)-1; \\ Michel Marcus, Nov 19 2017

a(n) = bitand(bitneg(n+1), n); \\ Ruud H.G. van Tol, Apr 05 2023

def A135481(n): return ~(n+1)&n # Chai Wah Wu, Jul 06 2022

a(n) = A006519(n+1) - 1. - R. J. Mathar, Feb 10 2016

a(0) = 0 prepended by Andrey Zabolotskiy, Oct 08 2019, based on Lothar Esser's contribution

A135517 a(n) = 2^(A091090(n)-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8

Offset: 0

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Also, a(n) = denominator(Euler(n, x) - Euler(n, 1)). - Observation from Peter Luschny, Aug 08 2017, proof from Vladimir Shevelev, Aug 13 2017

Also, a(n) = denominator(Euler(n,x) + Euler(n,0)). - Vladimir Shevelev, Aug 09 2017

Robert Israel, Table of n, a(n) for n = 0..10000
Vladimir Shevelev, Is A290646 = A135517?, Posting to Sequence Fans Mailing List, Aug 13 2017
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.

This is Guy Steele's sequence GS(2, 5) (see A135416).

Cf. A091090.

GS(2,5,200); # see A135416.
a := n -> `if`(n=1 or n mod 2 = 0, 1, 2*a(iquo(n,2))):
seq(a(n), n=0..103); # Peter Luschny, Aug 09 2017

b[n_] := b[n] = Which[n==0, 1, n==1, 1, EvenQ[n], 1, True, b[(n-1)/2] + 1]; a[n_] := 2^(b[n+1]-1); Array[a,103,0] (* Jean-François Alcover, Aug 12 2017 *)

a(n)=my(m=valuation(n+1,2)); 2^if(n>>m, m, m-1) \\ Charles R Greathouse IV, Aug 15 2017

def A135517(n): return (1<<(~(n+1)&n).bit_length()-(not n&(n+1))) if n else 1 # Chai Wah Wu, Sep 18 2024

For n >= 1, a(n) = 2^max_{odd k=1..n} (A007814(k+1) - t(n,k) - delta(n,k)), where delta(n,k) is the Kronecker symbol: delta(i,j) is 1 if i=j and 0 otherwise, and t(n,k) is the number of carries which appear in the addition of k and n-k in base 2. This allows us to answer in the affirmative the author's question (for a proof see Shevelev's link and its continuations). - Vladimir Shevelev, Aug 15 2017

Entry revised by N. J. A. Sloane, Aug 31 2017

A135523 a(n) = A007814(n) + A209229(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A135416, A007814, A036987.
This is Guy Steele's sequence GS(4, 1) (see A135416).
One less than A135560.

Maple
```
GS(4,1,200); [see A135416].
```

A135523(n) = (valuation(n,2)+(n && !bitand(n,n-1))); \\ Antti Karttunen, Sep 27 2018

def A135523(n): return (~n& n-1).bit_length()+int(not(n&-n)^n) # Chai Wah Wu, Jul 08 2022

G.f.: x + Sum_{k>=1} x^(2^k)*(1 + 1/(1 - x^(2^k))). - Ilya Gutkovskiy, Mar 30 2017

a(n) = A135560(n) - 1. Antti Karttunen, Sep 27 2018

A135534 a(1) = 1; for n>=1, a(2n) = A135561(n), a(2n+1) = 0.

Original entry on oeis.org

1, 3, 0, 7, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 63, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 127, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A135416.

This is Guy Steele's sequence GS(6, 1) (see A135416).

Maple
```
GS(6,1,200); [see A135416].
```

A135560(n) = { my(t=valuation(n, 2)); (t + (n==2^t) + 1); }; \\ From A135560
A135534(n) = if(1==n,1,if((n%2),0,((2^(A135560(n/2)))-1))); \\ Antti Karttunen, Sep 27 2018

def A135534(n): return 1 if n == 1 else 0 if n&1 else (1<<(m:=(~(k:=n>>1) & k-1)).bit_length()+int(m==k-1)+1)-1 # Chai Wah Wu, Jul 06 2022

A182660 a(2^(k+1)) = k; 0 everywhere else.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Sam Alexander, Nov 27 2010

nonn

A surjection N->N designed to spite a guesser who is trying to guess whether it's a surjection, using the following naive guessing method: Guess that (n0,...,nk) is a subsequence of a surjection iff it contains every natural less than log_2(k+1).
This sequence causes the would-be guesser to change his mind infinitely often.
a(0)=0. Assume a(0),...,a(n) have been defined.
If the above guesser guesses that (a(0),...,a(n)) IS the beginning of a surjective sequence, then let a(n+1)=0. Otherwise let a(n+1) be the least number not in (a(0),...,a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..65537
S. Alexander, On Guessing Whether A Sequence Has A Certain Property, arXiv:1011.6626 [math.LO], 2010-2012.
S. Alexander, On Guessing Whether A Sequence Has A Certain Property, J. Int. Seq. 14 (2011) # 11.4.4.

Cf. A082691, A135416, A209229.

[ exists(t){ k: k in [1..Ceiling(Log(n+1))] | n eq 2^(k+1) } select t else 0: n in [0..100] ];

A182660(n) = if(n<2,0,my(p = 0, k = isprimepower(n,&p)); if(2==p,k-1,0)); \\ Antti Karttunen, Jul 22 2018

A135521 a(n) = 2^(A091090(n)) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 63, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 63, 1, 3, 1, 7, 1

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

			From _Omar E. Pol_, Mar 11 2011: (Start)
Can be written as a triangle with 2^k entries on each row:
1,
1,3,
1,3,1,7,
1,3,1,7,1,3,1,15,
1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,
1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3, 1,7,1,3,1,63,
Last term of rows are 2^(k+1) - 1. It appears that the row sums give A001787.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A135416, A091090.
This is Guy Steele's sequence GS(2, 6) (see A135416).
Cf. A000225, A001787. - Omar E. Pol, Mar 11 2011

GS(2,6,200); [see A135416].
# Input n is the number of rows.
A135521_list := proc(n) local i,k,NimSum;
NimSum := proc(a,b) option remember; local i;
zip((x,y)->`if`(x<>y,1,0),convert(a,base,2),convert(b,base,2),0);
add(`if`(%[i]=1,2^(i-1),0),i=1..nops(%)) end:
seq(seq(NimSum(i,i+1),i=0..2^k-1),k=0..n) end:
A135521_list(5); # Peter Luschny, May 31 2011

Flatten[Table[BitXor[i, i + 1], {k, 0, 10}, {i, 0, -1 + 2^k}]] (* Peter Luschny, May 31 2011 *)

A091090(n) = { my(m=valuation(n+1, 2)); if(n>>m, m+1, max(m, 1)); }; \\ From A091090
A135521(n) = ((2^A091090(n))-1); \\ Antti Karttunen, Sep 27 2018

G.f. A(x) satisfies: A(x) = x/(1 - x) + 2*x*A(x^2). - Ilya Gutkovskiy, Dec 18 2019

A135540 a(n) = 2^(A000523(n) - A000120(n) + 2) - 1.

Original entry on oeis.org

1, 3, 1, 7, 3, 3, 1, 15, 7, 7, 3, 7, 3, 3, 1, 31, 15, 15, 7, 15, 7, 7, 3, 15, 7, 7, 3, 7, 3, 3, 1, 63, 31, 31, 15, 31, 15, 15, 7, 31, 15, 15, 7, 15, 7, 7, 3, 31, 15, 15, 7, 15, 7, 7, 3, 15, 7, 7, 3, 7, 3, 3, 1, 127, 63, 63, 31, 63, 31, 31, 15, 63, 31, 31, 15, 31, 15, 15, 7, 63, 31, 31, 15, 31, 15

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A135416.

This is Guy Steele's sequence GS(6, 3) (see A135416).

Maple
```
GS(6,3,200); [see A135416].
```

Table[2^(Floor[Log[2, n]] - (n - Sum[Floor[n/2^k], {k, 1, n}]) + 2) - 1, {n,1,25}] (* G. C. Greubel, Oct 18 2016 *)

a(n) = 2^(floor(log_2(n)) - (n - Sum_{k=1,..,n}[floor(n/2^k)]) + 2) - 1. - G. C. Greubel, Oct 18 2016

cp's OEIS Frontend

A091090 In binary representation: number of editing steps (delete, insert, or substitute) to transform n into n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Haskell

Haskell

Maple

Mathematica

PARI

Python

Formula

A008687 Number of 1's in 2's complement representation of -n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A135528 1, then repeat 1,0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A135481 a(n) = 2^A007814(n+1) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Julia

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A135517 a(n) = 2^(A091090(n)-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A135523 a(n) = A007814(n) + A209229(n).

Views

Author

Keywords

Links

Crossrefs

Programs