

A240542


Numbers a(n) that are the coordinates of the midpoints of the (rotated) Dyck paths from (0, n) to (n, 0) defined by A237593. Also the alternating row sums of A235791.


39



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

1,2


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*log2)/(k*log2) <= 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(n1), 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 oddindexed terms of the nth row of the triangle A237591.  Omar E. Pol, Jun 20 2018


LINKS

Table of n, a(n) for n=1..73.


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


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}]


PROG

(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 xrange(1, int(math.floor(1/2 + 1/2*sqrt(8*n + 1))) + 1)])
print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, Apr 21 2017


CROSSREFS

Cf. A028982, A067742, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A259179, A279387.
Sequence in context: A038810 A178503 A211275 * A325391 A179254 A304430
Adjacent sequences: A240539 A240540 A240541 * A240543 A240544 A240545


KEYWORD

nonn


AUTHOR

Hartmut F. W. Hoft, Apr 07 2014


EXTENSIONS

More terms from Omar E. Pol, Apr 16 2014


STATUS

approved



