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A337765
Number of addition triangles with apex n where all rows are weakly increasing.
4
1, 2, 2, 4, 4, 5, 6, 9, 9, 11, 12, 15, 16, 18, 20, 26, 27, 29, 32, 37, 39, 43, 47, 53, 55, 60, 65, 72, 75, 80, 88, 99, 102, 108, 114, 125, 132, 141, 148, 159, 166, 176, 187, 200, 206, 218, 232, 249, 257, 268, 282, 301, 313, 327, 340, 360, 374, 393, 410, 429, 444, 465, 487, 516, 530, 550
OFFSET
1,2
COMMENTS
An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.
If the bottom row are weakly increasing, then every rows are weakly increasing.
5
2<=3
1<=1<=2
LINKS
EXAMPLE
For n = 5:
5
2,3 5 5
1,1,2 1,4 2,3 5
For n = 6:
6
2,4 6 6 6
1,1,3 1,5 2,4 3,3 6
For n = 7:
7 7
2,5 3,4 7 7 7
1,1,4 1,2,2 1,6 2,5 3,4 7
For n = 8:
8
4,4 8 8 8
2,2,2, 2,6 3,5 4,4 8 8 8 8
1,1,1,1 1,1,5 1,2,3 2,2,2 1,7 2,6 3,5 4,4 8
For n = 9:
9
4,5 9 9 9
2,2,3, 2,7 3,6 4,5 9 9 9 9
1,1,1,2 1,1,6 1,2,4 2,2,3 1,8 2,7 3,6 4,5 9
PROG
(Ruby)
def A(n)
f_ary = [[n]]
cnt = 1
while f_ary.size > 0
b_ary = []
f_ary.each{|i|
s = i.size
(1..i[0] - 1).each{|j|
a = [j]
(0..s - 1).each{|k|
num = i[k] - a[k]
if num > 0
a << num
else
break
end
}
b_ary << a if a.size == s + 1 && a == a.sort
}
}
f_ary = b_ary
cnt += f_ary.size
end
cnt
end
def A337765(n)
(1..n).map{|i| A(i)}
end
p A337765(50)
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 19 2020
STATUS
approved