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A240542
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The midpoint of the (rotated) Dyck path from (0, n) to (n, 0) defined by A237593 has coordinates (a(n), a(n)). Also a(n) is the alternating sum of the n-th row of A235791.
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54
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1, 2, 2, 3, 3, 5, 5, 6, 7, 7, 7, 9, 9, 9, 11, 12, 12, 13, 13, 15, 15, 15, 15, 17, 18, 18, 18, 20, 20, 22, 22, 23, 23, 23, 25, 26, 26, 26, 26, 28, 28, 30, 30, 30, 32, 32, 32, 34, 35, 36, 36, 36, 36, 38, 38, 40, 40, 40, 40, 42, 42, 42, 44, 45, 45, 47, 47, 47, 47, 49, 49, 52, 52
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OFFSET
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1,2
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COMMENTS
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The sequence is closely related to the alternating harmonic series.
Its asymptotic behavior is lim_{k -> infinity} a(k)/k = log 2. The relative error is abs(a(k) - k*log(2))/(k*log(2)) <= 2/sqrt(k).
Conjecture 1: the sequence of first positions of the alternating runs of odd and even numbers in a(k) equals A028982. Example: the positions in (1),(2),2,(3),3,5,5,(6),(7),7,7,9,9,9,11,(12),12,(13),13,15,... are 1,2,4,8,9,16,18,... Checked with a Mathematica function through a(1000000).
Conjecture 2: The sequence of differences a(n) - a(n-1), for n>=1, appears to be equal to A067742(n), the sequence of middle divisors of n; as an empty sum, a(0) = 0, (which was conjectured by Michel Marcus in the entry A237593). Checked with the two respective Mathematica functions up to n=100000. - Hartmut F. W. Hoft, Jul 17 2014
Conjecture 3: a(n) is also the difference between the total number of partitions of all positive integers <= n into an odd number of consecutive parts, and the total number of partitions of all positive integers <= n into an even number of consecutive parts. - Omar E. Pol, Apr 28 2017
Conjecture 4: a(n) is also the total number of central subparts of all symmetric representations of sigma of all positive integers <= n. - Omar E. Pol, Apr 29 2017
a(n) is also the sum of the odd-indexed terms of the n-th row of the triangle A237591. - Omar E. Pol, Jun 20 2018
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LINKS
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FORMULA
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EXAMPLE
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Illustration of initial terms in two ways in accordance with the sum of the odd-indexed terms of the rows of A237591:
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n a(n) _ _
1 1 _|_| |_|_
2 2 _|_ _| |_ _|
3 2 _|_ _| |_ _|_
4 3 _|_ _ _| |_ _ _|
5 3 _|_ _ _| _ |_ _ _|_ _
6 5 _|_ _ _ _| |_| |_ _ _ _|_|
7 5 _|_ _ _ _| |_| |_ _ _ _|_|_
8 6 _|_ _ _ _ _| _|_| |_ _ _ _ _|_|_
9 7 _|_ _ _ _ _| |_ _| |_ _ _ _ _|_ _|
10 7 _|_ _ _ _ _ _| |_| |_ _ _ _ _ _|_|
11 7 _|_ _ _ _ _ _| _|_| |_ _ _ _ _ _|_|_ _
12 9 _|_ _ _ _ _ _ _| |_ _| |_ _ _ _ _ _ _|_ _|
13 9 _|_ _ _ _ _ _ _| |_ _| |_ _ _ _ _ _ _|_ _|
14 9 _|_ _ _ _ _ _ _ _| _|_| _ |_ _ _ _ _ _ _ _|_|_ _
15 11 _|_ _ _ _ _ _ _ _| |_ _| |_| |_ _ _ _ _ _ _ _|_ _|_|_
16 12 |_ _ _ _ _ _ _ _ _| |_ _| |_| |_ _ _ _ _ _ _ _ _|_ _|_|
...
Figure 1. Figure 2.
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Figure 1 shows the illustration of initial terms taken from the isosceles triangle of A237593. For n = 16 there are (9 + 2 + 1) = 12 cells in the 16th row of the diagram, so a(16) = 12.
Figure 2 shows the illustration of initial terms taken from an octant of the pyramid described in A244050 and A245092. For n = 16 there are (9 + 2 + 1) = 12 cells in the 16th row of the diagram, so a(16) = 12.
Note that if we fold each level (or row) of that isosceles triangle of A237593 we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n).
(End).
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MATHEMATICA
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a[n_] := Sum[(-1)^(k + 1) Ceiling[(n + 1)/k - (k + 1)/2], {k, 1, Floor[-1/2 + 1/2 Sqrt[8 n + 1]]}]; Table[a[n], {n, 40}]
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PROG
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(PARI) a(n) = sum(k=1, floor(-1/2 + 1/2*sqrt(8*n + 1)), (-1)^(k + 1)*ceil((n + 1)/k - (k + 1)/2)); \\ Indranil Ghosh, Apr 21 2017
(Python)
from sympy import sqrt
import math
def a(n): return sum((-1)**(k + 1) * int(math.ceil((n + 1)/k - (k + 1)/2)) for k in range(1, int(math.floor(-1/2 + 1/2*sqrt(8*n + 1))) + 1))
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CROSSREFS
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Cf. A028982, A067742, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A259176, A259179, A279387, A286001, A322141, A338204, A338758.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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