cp's OEIS Frontend

A246878 a(0) = 1, then a(n) = sum(a(k), k = floor(log_2(n)) .. n - 1).

%I A246878 #20 May 04 2016 21:39:25
%S A246878 1,1,1,2,3,6,12,24,47,94,188,376,752,1504,3008,6016,12030,24060,48120,
%T A246878 96240,192480,384960,769920,1539840,3079680,6159360,12318720,24637440,
%U A246878 49274880,98549760,197099520,394199040,788398077
%N A246878 a(0) = 1, then a(n) = sum(a(k), k = floor(log_2(n)) .. n - 1).
%C A246878 a(n) = Theta(2^n), and more precisely, for n >= 4, (13/16)*(3/16)2^n <= a(n) <= (3/16)*2^n.
%C A246878 Indeed, from the formula, one gets a(n) <= (3/16)*2^n, and injecting this in the formula, one gets a(n) >= 2*a(n - 1) - (3/32)*n. Then by induction, and using the formula sum(k*2^k, k = 1 .. n) = (n - 1)*2^(n + 1) + 2, one obtains a(n) >= (13/16)*(3/16)2^n + (3/32)*n + (3/8).
%H A246878 Reinhard Zumkeller, <a href="/A246878/b246878.txt">Table of n, a(n) for n = 0..1000</a>
%H A246878 W. Kuszmaul, <a href="http://arxiv.org/abs/1509.08216">Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations</a>, arXiv preprint arXiv:1509.08216, 2015
%F A246878 If n >= 1 is not a power of 2, then a(n) = 2*a(n - 1), and if k >= 1, then a(2^k) = 2*a(2^k - 1) - a(k - 1).
%e A246878 a(2) = a(1) = a(0) = 1.
%e A246878 a(3) = a(2) + a(1) = 2.
%e A246878 a(4) = a(3) + a(2) = 3.
%e A246878 a(5) = a(4) + a(3) + a(2) = 6.
%o A246878 (Haskell)
%o A246878 import Data.List (genericDrop)
%o A246878 a246878 n = a246878_list !! n
%o A246878 a246878_list = 1 : f [1] a000523_list where
%o A246878    f xs (k:ks) = y : f (xs ++ [y]) ks where y = sum $ genericDrop k xs
%o A246878 -- _Reinhard Zumkeller_, Sep 16 2014
%Y A246878 Cf. A000523.
%K A246878 nonn,easy
%O A246878 0,4
%A A246878 _Benoit Jubin_, Sep 06 2014