A003314 Binary entropy function: a(1)=0; for n > 1, a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }. (Formerly M1345)

0, 2, 5, 8, 12, 16, 20, 24, 29, 34, 39, 44, 49, 54, 59, 64, 70, 76, 82, 88, 94, 100, 106, 112, 118, 124, 130, 136, 142, 148, 154, 160, 167, 174, 181, 188, 195, 202, 209, 216, 223, 230, 237, 244, 251, 258, 265, 272, 279, 286, 293, 300, 307, 314, 321, 328, 335

Offset

1,2

Comments

Morris gives many other interesting properties of this function.

a(n) is a convex function of n. (See the Morris reference.)

References

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.4.9, Eq. (19). p. 374.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Indranil Ghosh, Table of n, a(n) for n = 1..32768 (terms 1..1000 from T. D. Noe)

Mareike Fischer, Extremal values of the Sackin balance index for rooted binary trees, arXiv:1801.10418 [q-bio.PE], 2018.

Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 4.

D. Chistikov, S. Iván, A. Lubiw, and J. Shallit, Fractional coverings, greedy coverings, and rectifier networks, arXiv preprint arXiv:1509.07588 [cs.CC], 2015-2016.

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.

D. Knuth, Letter to N. J. A. Sloane, date unknown

R. Morris, Some theorems on sorting, SIAM J. Appl. Math., 17 (1969), 1-6.

Formula

a(1) = 0; a(n) = n + a([n/2]) + a(n-[n/2]). (See the Morris reference.)

a(n) = A001855(n)+n-1. - Michael Somos Feb 07 2004

a(n) = n + a(floor(n/2)) + a(ceiling(n/2)) = n*floor(log_2(4n))-2^floor(log_2(2n)) = A033156(n) - n = n*A070941(n) - A062383(n). - Henry Bottomley, Jul 03 2002

a(1) = 0 and for n>1: a(n) = a(n-1) + A070941(2*n-1). Also a(n) = A123753(n-1) - 1. - Reinhard Zumkeller, Oct 12 2006

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

Example

a(6) = 6 + min {1+12, 2+8, 5+5} = 6 + 10 = 16.

Maple

A003314 := proc(n) local i, j; option remember; if n<=2 then n elif n=3 then 5 else j := 10^10; for i from 1 to n-1 do if A003314(i)+A003314(n-i) < j then j := A003314(i)+A003314(n-i); fi; od; n+j; fi; end;

Mathematica

a[1] = 0; a[n_] := If[OddQ[n], n + a[(n-1)/2 + 1] + a[(n-1)/2], 2*(n/2 + a[n/2])];
Table[a[n], {n, 1, 57}] (* Jean-François Alcover, Oct 15 2012 *)
a[n_] := n + n IntegerLength[n, 2] - 2^IntegerLength[n, 2];
Table[a[n], {n, 1, 57}] (* Peter Luschny, Dec 02 2017 *)

Prog

(PARI) a(n)=if(n<2, 0, n+a(n\2)+a((n+1)\2))
(PARI) a(n)=local(m); if(n<2, 0, m=length(binary(n-1)); n*m-2^m+n)
(Haskell)
a003314 n = a003314_list !! (n-1)
a003314_list = 0 : f [0] [2..] where
   f vs (w:ws) = y : f (y:vs) ws where
     y = w + minimum (zipWith (+) vs $ reverse vs)
-- Reinhard Zumkeller, Aug 13 2013
(Python)
def A003314(n):
    return n*int(math.log(4*n, 2))-2**int(math.log(2*n, 2)) # Indranil Ghosh, Feb 03 2017
(Python)
def A003314(n):
    s, i, z = n-1, n-1, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A003314(n) for n in range(1, 58)]) # Peter Luschny, Nov 30 2017
(Python)
def A003314(n): return n*(1+(m:=(n-1).bit_length()))-(1<<m) # Chai Wah Wu, Mar 29 2023

Crossrefs

Cf. A054248, A070941, A123753.

Sequence in context: A228153 A228152 A061717 * A228155 A070977 A134925

Adjacent sequences: A003311 A003312 A003313 * A003315 A003316 A003317

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved