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A285356
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Numbers n such that the entries in the n-th row of the irregular triangle A237591 are in nonincreasing order.
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4
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 36, 37, 38, 40, 41, 45, 46, 47, 48, 51, 55, 57, 58, 59, 66, 67, 70, 71, 78, 79, 80, 84, 92, 93, 94, 108, 109, 120, 136, 137, 155
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OFFSET
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1,2
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COMMENTS
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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.
(4) For all n > 231 = A066370(2), the position of the rightmost inversion in the n-th row is given by the formula r(n) - r( Binomial( r(n) + 2, 2) - 1 - n); tested through 2500000. Expressed in terms of A-numbers the formula is: A003056(n) - A003056(A000217(A003056(n) + 1) - 1 - n).
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LINKS
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EXAMPLE
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19 is in the sequence since row 19 in A237591 is 10, 4, 2, 2, 1.
20 is not in the sequence since row 20 in A237591 is 11, 4, 2, 1, 2.
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MATHEMATICA
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(* 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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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