OFFSET
1,2
COMMENTS
For the numbers n in the sequence the lengths of the steps in the (first half of the) associated Dyck path of A237593 are nonincreasing.
Conjectures:
(1) The sequence consists of the 59 numbers listed above; tested through 5000000.
(2) The values f(n,k) in the n-th row of triangle A237591 are either 1 or 2 for all k with ceiling((sqrt(4*n+1)-1)/2) <= k <= floor((sqrt(8*n+1)-1)/2) = r(n), the length of the n-th row, though the lower bound need not be minimal; tested through 2500000.
(3) For every n > 155 there is an inversion 1 = f(n,k-1) < f(n,k) = 2 where k >= ceiling((sqrt(4*n+1)-1)/2, except the inversions for n = 174 at k = 12 and for n = 231 at k = 14; tested through 2500000.
LINKS
EXAMPLE
MATHEMATICA
(* functions row[] and f[] are defined in A237591 *)
nonincreasingQ[n_] := Module[{i=2, b=row[n], good=True}, While[good && i<=b, good=good && (f[n, i]<=f[n, i-1]); i++]; good]
a285356[m_, n_] := Module[{i, sols={}}, For[i=m, i<=n, i++, If[nonincreasingQ[i], AppendTo[sols, i]]]; sols]
a285356[1, 200] (* data *)
PROG
(Python)
import math
from sympy import sqrt
def T(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2)) - int(math.ceil((n + 1)/(k + 1) - (k + 2)/2))
def isok(n):
l = [T(n, k) for k in range(1, int(math.floor((sqrt(8*n + 1) - 1)/2)) + 1)]
for i in range(len(l) - 1):
if l[i + 1] > l[i]: return 0
return 1
print([n for n in range(1, 156) if isok(n)]) # Indranil Ghosh, Apr 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Hartmut F. W. Hoft, Apr 17 2017
STATUS
approved