cp's OEIS Frontend

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

%I A062050 #91 Jan 07 2025 05:44:39
%S A062050 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,
%T A062050 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,
%U A062050 27,28,29,30,31,32,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17
%N A062050 n-th chunk consists of the numbers 1, ..., 2^n.
%C A062050 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
%C A062050 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
%H A062050 Reinhard Zumkeller, <a href="/A062050/b062050.txt">Table of n, a(n) for n = 1..10000</a>
%H A062050 Paul Barry, <a href="https://arxiv.org/abs/2107.00442">Conjectures and results on some generalized Rueppel sequences</a>, arXiv:2107.00442 [math.CO], 2021.
%H A062050 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences with (relatively) simple ordinary generating functions</a>, 2004.
%H A062050 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>. [ps file]
%H A062050 Ralf Stephan, <a href="/A062050/a062050.pdf">Table of generating functions</a>. [pdf file]
%F A062050 a(n) = A053645(n) + 1.
%F A062050 a(n) = n - msb(n) + 1 (where msb(n) = A053644(n)).
%F A062050 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
%F A062050 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
%F A062050 a(1) = 1, a(2*n) = 2*a(n) - 1, a(2*n+1) = 2*a(n). - _Ralf Stephan_, Oct 06 2003
%F A062050 A005836(a(n+1)) = A107681(n). - _Reinhard Zumkeller_, May 20 2005
%F A062050 a(n) = if n < 2 then n else 2*a(floor(n/2)) - 1 + n mod 2. - _Reinhard Zumkeller_, May 07 2012
%F A062050 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
%F A062050 For n > 0: a(n) = (A006257(n) + 1) / 2. - _Frank Hollstein_, Oct 25 2021
%e A062050 From _Omar E. Pol_, Aug 31 2013: (Start)
%e A062050 Written as irregular triangle with row lengths A000079:
%e A062050   1;
%e A062050   1, 2;
%e A062050   1, 2, 3, 4;
%e A062050   1, 2, 3, 4, 5, 6, 7, 8;
%e A062050   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
%e A062050   ...
%e A062050 Row sums give A007582.
%e A062050 (End)
%p A062050 A062050 := proc(n) option remember; if n < 4 then return [1, 1, 2][n] fi;
%p A062050 2*A062050(floor(n/2)) + irem(n,2) - 1 end:
%p A062050 seq(A062050(n), n=1..89); # _Peter Luschny_, Apr 27 2020
%t A062050 Flatten[Table[Range[2^n],{n,0,6}]] (* _Harvey P. Dale_, Oct 12 2015 *)
%o A062050 (PARI) a(n)=floor(n+1-2^logint(n,2))
%o A062050 (PARI) a(n)= n - 1<<logint(n,2) + 1; \\ _Ruud H.G. van Tol_, Dec 13 2024
%o A062050 (Haskell)
%o A062050 a062050 n = if n < 2 then n else 2 * a062050 n' + m - 1
%o A062050             where (n',m) = divMod n 2
%o A062050 -- _Reinhard Zumkeller_, May 07 2012
%o A062050 (Python)
%o A062050 def A062050(n): return n-(1<<n.bit_length()-1)+1 # _Chai Wah Wu_, Jan 22 2023
%Y A062050 Cf. A053644, A053645.
%Y A062050 Cf. A092754.
%K A062050 nonn,easy
%O A062050 1,3
%A A062050 _Marc LeBrun_, Jun 30 2001