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A054248
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Binary entropy: a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }.
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4
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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REFERENCES
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D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 374.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..10000
Hsien-Kuei Hwang, S. Janson, 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, 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.
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FORMULA
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a(n) = A123753(n-1) - (n-1) mod 2. - Peter Luschny, Nov 30 2017
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A003314, A123753.
Sequence in context: A168247 A229056 A186703 * A038108 A294862 A087327
Adjacent sequences: A054245 A054246 A054247 * A054249 A054250 A054251
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, May 04 2000
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STATUS
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approved
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