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A062050 n-th chunk consists of the numbers 1, ..., 2^n. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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]

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

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

EXAMPLE

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)

MAPLE

a := proc(n) option remember; if n < 4 then return [1, 1, 2][n] fi;

2*a(floor(n/2)) + irem(n, 2) - 1 end: seq(a(n), n=1..89); # Peter Luschny, Apr 27 2020

MATHEMATICA

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

PROG

(PARI) a(n)=floor(n+1-2^floor(log(n+1-10^-27)/log(2)))

(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

CROSSREFS

Cf. A053644, A053645.

Cf. A092754.

Sequence in context: A194103 A074294 A168265 * A233782 A233972 A169778

Adjacent sequences:  A062047 A062048 A062049 * A062051 A062052 A062053

KEYWORD

nonn

AUTHOR

Marc LeBrun, Jun 30 2001

STATUS

approved

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Last modified August 5 22:41 EDT 2021. Contains 346488 sequences. (Running on oeis4.)