A240542 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.

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

Offset: 1

Author

Hartmut F. W. Hoft, Apr 07 2014

Keywords

• nonn

Comments

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).

See A235791, A237591 and A237593 for additional formulas and properties.

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

The number of occurrences of n is A259179(n). - Omar E. Pol, Dec 11 2016

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

a(n) is the total number of middle divisors of all positive integers <= n (after Michel Marcus's conjecture in A237593). - Omar E. Pol, Aug 18 2021

Examples

			From _Omar E. Pol_, Dec 22 2020: (Start)
Illustration of initial terms in two ways in accordance with the sum of the odd-indexed terms of the rows of A237591:
.
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.
.
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).
		

Crossrefs

Cf. A028982, A067742, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A259176, A259179, A279387, A286001, A322141, A338204, A338758.

Programs

Mathematica

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}]

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))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 21 2017

Formula

a(n) = Sum_{k = 1..A003056(n)} (-1)^(k+1) A235791(n,k).

a(n) = A285901(n) - A285902(n), assuming the conjecture 3. - Omar E. Pol, Feb 15 2018

a(n) = n - A322141(n). - Omar E. Pol, Dec 22 2020

Extensions

More terms from Omar E. Pol, Apr 16 2014

Definition edited by N. J. A. Sloane, Dec 20 2020