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A033156
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a(1) = 1; for m >= 2, a(n) = a(n-1) + floor(a(n-1)/(n-1)) + 2.
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6
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1, 4, 8, 12, 17, 22, 27, 32, 38, 44, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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LINKS
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Harvey P. Dale, Table of n, a(n) for n = 1..1000
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.
Haomin Li, Computing a Basis for an Integer Lattice, Master's Thesis, Univ. of Waterloo (Ontario, Canada 2022).
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 200), 559-564, Th. 3.1.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
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FORMULA
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a(n) = n*(floor(log_2 n) + 3) - 2^((floor (log_2 n)) + 1).
a(n) = n + a(floor(n/2)) + a(ceiling(n/2)) = n + min{a(k) + a(n-k):0 < k < n} = n + A003314(n). - Henry Bottomley, Jul 03 2002
A001855(n) + 2n-1. a(n) = b(n)+1 with b(0)=0, b(2n) = b(n) + b(n-1) + 2n + 2, b(2n+1) = 2b(n) + 2n + 3. - Ralf Stephan, Oct 24 2003
a(n) = A123753(n-1) + n - 1. - Peter Luschny, Nov 30 2017
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MAPLE
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A033156 := proc(n) option remember; if n=1 then 1 else A033156(n-1)+floor(A033156(n-1)/(n-1))+2; fi; end;
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MATHEMATICA
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a[n_] := n (2 + IntegerLength[n, 2]) - 2^IntegerLength[n, 2];
Table[a[n], {n, 1, 59}] (* Peter Luschny, Dec 02 2017 *)
nxt[{n_, a_}]:={n+1, a+Floor[a/n]+2}; NestList[nxt, {1, 1}, 60][[All, 2]] (* Harvey P. Dale, Nov 03 2020 *)
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PROG
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(Python)
def A033156(n):
s, i, z = 2*n-1, n-1, 1
while 0 <= i: s += i; i -= z; z += z
return s
print([A033156(n) for n in range(1, 60)]) # Peter Luschny, Nov 30 2017
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CROSSREFS
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Cf. A123753.
Sequence in context: A002004 A311537 A311538 * A311539 A311540 A311541
Adjacent sequences: A033153 A033154 A033155 * A033157 A033158 A033159
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Jun 05 2002
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STATUS
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approved
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