A062050 n-th chunk consists of the numbers 1, ..., 2^n.

1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

Offset: 1

Author

Marc LeBrun, Jun 30 2001

Keywords

• nonn
• easy

Comments

a(k) is the distance between k and the largest power of 2 not exceeding k, where k = n + 1. [Consider the sequence of even numbers <= k; after sending the first term to the last position delete all odd-indexed terms; the final term that remains after iterating the process is the a(k)-th even number.] - Lekraj Beedassy, May 26 2005

Triangle read by rows in which row n lists the first 2^(n-1) positive integers, n >= 1; see the example. - Omar E. Pol, Sep 10 2013

Examples

			From _Omar E. Pol_, Aug 31 2013: (Start)
Written as irregular triangle with row lengths A000079:
  1;
  1, 2;
  1, 2, 3, 4;
  1, 2, 3, 4, 5, 6, 7, 8;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
  ...
Row sums give A007582.
(End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
• Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
• Ralf Stephan, Table of generating functions. [ps file]
• Ralf Stephan, Table of generating functions. [pdf file]

Crossrefs

Cf. A053644, A053645.

Cf. A092754.

Programs

Haskell

a062050 n = if n < 2 then n else 2 * a062050 n' + m - 1
            where (n',m) = divMod n 2
-- Reinhard Zumkeller, May 07 2012

Maple

A062050 := proc(n) option remember; if n < 4 then return [1, 1, 2][n] fi;
2*A062050(floor(n/2)) + irem(n,2) - 1 end:
seq(A062050(n), n=1..89); # Peter Luschny, Apr 27 2020

Mathematica

Flatten[Table[Range[2^n],{n,0,6}]] (* Harvey P. Dale, Oct 12 2015 *)

PARI

a(n)=floor(n+1-2^logint(n,2))

PARI

a(n)= n - 1<Ruud H.G. van Tol, Dec 13 2024

Python

def A062050(n): return n-(1<Chai Wah Wu, Jan 22 2023

Formula

a(n) = A053645(n) + 1.

a(n) = n - msb(n) + 1 (where msb(n) = A053644(n)).

a(n) = 1 + n - 2^floor(log(n)/log(2)). - Benoit Cloitre, Feb 06 2003; corrected by Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 25 2008

G.f.: 1/(1-x) * ((1-x+x^2)/(1-x) - Sum_{k>=1} 2^(k-1)*x^(2^k)). - Ralf Stephan, Apr 18 2003

a(1) = 1, a(2*n) = 2*a(n) - 1, a(2*n+1) = 2*a(n). - Ralf Stephan, Oct 06 2003

A005836(a(n+1)) = A107681(n). - Reinhard Zumkeller, May 20 2005

a(n) = if n < 2 then n else 2*a(floor(n/2)) - 1 + n mod 2. - Reinhard Zumkeller, May 07 2012

Without the constant 1, Ralf Stephan's g.f. becomes A(x) = x/(1-x)^2 - (1/(1-x)) * Sum_{k>=1} 2^(k-1)*x^(2^k) and satisfies the functional equation A(x) - 2*(1+x)*A(x^2) = x*(1 - x - x^2)/(1 - x^2). - Petros Hadjicostas, Apr 27 2020

For n > 0: a(n) = (A006257(n) + 1) / 2. - Frank Hollstein, Oct 25 2021