%I #23 Nov 21 2021 08:13:57

%S 1,1,2,2,3,3,4,5,6,6,8,9,10,11,13,14,16,17,19,22,24,25,28,31,33,35,39,

%T 43,46,48,52,57,60,63,69,75,78,82,88,94,99,104,111,119,124,129,137,

%U 147,153,160,169,179,187,194,204,216,224,233,246,259,267,277,292,308,318,329,343,361

%N Number of addition triangles with apex n where all rows are strongly increasing.

%C An addition triangle has any finite sequence of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.

%C If the bottom row is strongly increasing, then every row is strongly increasing.

%C 8

%C 3<5

%C 1<2<3

%H Seiichi Manyama, <a href="/A337766/b337766.txt">Table of n, a(n) for n = 1..500</a>

%e For n = 5:

%e 5 5

%e 1,4 2,3 5

%e For n = 6:

%e 6 6

%e 1,5 2,4 6

%e For n = 7:

%e 7 7 7

%e 1,6 2,5 3,4 7

%e For n = 8:

%e 8

%e 3,5 8 8 8

%e 1,2,3 1,7 2,6 3,5 8

%e For n = 9:

%e 9

%e 3,6 9 9 9 9

%e 1,2,4 1,8 2,7 3,6 4,5 9

%o (Ruby)

%o def A(n)

%o f_ary = [[n]]

%o cnt = 1

%o while f_ary.size > 0

%o b_ary = []

%o f_ary.each{|i|

%o s = i.size

%o (1..i[0] - 1).each{|j|

%o a = [j]

%o (0..s - 1).each{|k|

%o num = i[k] - a[k]

%o if num > 0

%o a << num

%o else

%o break

%o end

%o }

%o b_ary << a if a.size == s + 1 && a == a.uniq.sort

%o }

%o }

%o f_ary = b_ary

%o cnt += f_ary.size

%o end

%o cnt

%o end

%o def A337766(n)

%o (1..n).map{|i| A(i)}

%o end

%o p A337766(50)

%Y Equivalent sequences with different restrictions on rows: A062684 (none, except terms are positive), A062896 (not a reversal of a counted row), A337765 (weakly increasing).

%Y Cf. A346523.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Sep 19 2020