A054248 Binary entropy: a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }.

1, 2, 6, 8, 13, 16, 21, 24, 30, 34, 40, 44, 50, 54, 60, 64, 71, 76, 83, 88, 95, 100, 107, 112, 119, 124, 131, 136, 143, 148, 155, 160, 168, 174, 182, 188, 196, 202, 210, 216, 224, 230, 238, 244, 252, 258, 266, 272, 280, 286, 294, 300, 308, 314, 322, 328, 336

Offset: 1

Author

N. J. A. Sloane, May 04 2000

Keywords

• nonn

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.

Crossrefs

Cf. A003314, A123753.

Programs

Maple

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;
# second Maple program:
a:= proc(n) option remember; `if`(n<3, n,
      n + min(seq(a(k)+a(n-k), k=1..n/2)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 29 2015

Mathematica

a[n_] := n + n IntegerLength[n, 2] - 2^IntegerLength[n, 2] + Mod[n, 2];
Table[a[n], {n, 1, 54}] (* Peter Luschny, Dec 02 2017 *)

Python

def A054248(n):
    s, i, z = n - (n-1) % 2, n-1, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A054248(n) for n in range(1, 55)]) # Peter Luschny, Nov 30 2017

Python

def A054248(n): return n*(1+(m:=(n-1).bit_length()))-(1<Chai Wah Wu, Mar 29 2023

Formula

a(n) = A123753(n-1) - (n-1) mod 2. - Peter Luschny, Nov 30 2017