login
Binary entropy: a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }.
4

%I #30 Mar 30 2023 01:59:39

%S 1,2,6,8,13,16,21,24,30,34,40,44,50,54,60,64,71,76,83,88,95,100,107,

%T 112,119,124,131,136,143,148,155,160,168,174,182,188,196,202,210,216,

%U 224,230,238,244,252,258,266,272,280,286,294,300,308,314,322,328,336

%N Binary entropy: a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }.

%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 374.

%H Alois P. Heinz, <a href="/A054248/b054248.txt">Table of n, a(n) for n = 1..10000</a>

%H Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications</a>, Preprint 2016.

%H Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="https://doi.org/10.1145/3127585">Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications</a>, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.

%F a(n) = A123753(n-1) - (n-1) mod 2. - _Peter Luschny_, Nov 30 2017

%p A054248 := proc(n) local i,j; option remember; if n<=2 then n else j := 10^10; for i from 1 to n-1 do if A054248(i)+A054248(n-i) < j then j := A054248(i)+A054248(n-i); fi; od; n+j; fi; end;

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<3, n,

%p n + min(seq(a(k)+a(n-k), k=1..n/2)))

%p end:

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Aug 29 2015

%t a[n_] := n + n IntegerLength[n, 2] - 2^IntegerLength[n, 2] + Mod[n, 2];

%t Table[a[n], {n, 1, 54}] (* _Peter Luschny_, Dec 02 2017 *)

%o (Python)

%o def A054248(n):

%o s, i, z = n - (n-1) % 2, n-1, 1

%o while 0 <= i: s += i; i -= z; z += z

%o return s

%o print([A054248(n) for n in range(1, 55)]) # _Peter Luschny_, Nov 30 2017

%o (Python)

%o def A054248(n): return n*(1+(m:=(n-1).bit_length()))-(1<<m)+(n&1) # _Chai Wah Wu_, Mar 29 2023

%Y Cf. A003314, A123753.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, May 04 2000