OFFSET
1,3
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
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]
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
For n > 0: a(n) = (A006257(n) + 1) / 2. - Frank Hollstein, Oct 25 2021
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
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 *)
PROG
(PARI) a(n)=floor(n+1-2^floor(log(n+1-10^-27)/log(2)))
(PARI) a(n)= n - 1<<logint(n, 2) + 1; \\ Ruud H.G. van Tol, Dec 13 2024
(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
(Python)
def A062050(n): return n-(1<<n.bit_length()-1)+1 # Chai Wah Wu, Jan 22 2023
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Marc LeBrun, Jun 30 2001
STATUS
approved