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A337765 Number of addition triangles with apex n where all rows are weakly increasing.

%I A337765 #24 Sep 20 2020 04:58:16
%S A337765 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,
%T A337765 65,72,75,80,88,99,102,108,114,125,132,141,148,159,166,176,187,200,
%U A337765 206,218,232,249,257,268,282,301,313,327,340,360,374,393,410,429,444,465,487,516,530,550
%N A337765 Number of addition triangles with apex n where all rows are weakly increasing.
%C A337765 An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.
%C A337765 If the bottom row are weakly increasing, then every rows are weakly increasing.
%C A337765      5
%C A337765    2<=3
%C A337765   1<=1<=2
%H A337765 Seiichi Manyama, <a href="/A337765/b337765.txt">Table of n, a(n) for n = 1..500</a>
%e A337765 For n = 5:
%e A337765     5
%e A337765    2,3     5     5
%e A337765   1,1,2   1,4   2,3   5
%e A337765 For n = 6:
%e A337765     6
%e A337765    2,4     6     6     6
%e A337765   1,1,3   1,5   2,4   3,3   6
%e A337765 For n = 7:
%e A337765     7       7
%e A337765    2,5     3,4     7     7     7
%e A337765   1,1,4   1,2,2   1,6   2,5   3,4   7
%e A337765 For n = 8:
%e A337765      8
%e A337765     4,4       8       8       8
%e A337765    2,2,2,    2,6     3,5     4,4     8     8     8     8
%e A337765   1,1,1,1   1,1,5   1,2,3   2,2,2   1,7   2,6   3,5   4,4   8
%e A337765 For n = 9:
%e A337765      9
%e A337765     4,5       9       9       9
%e A337765    2,2,3,    2,7     3,6     4,5     9     9     9     9
%e A337765   1,1,1,2   1,1,6   1,2,4   2,2,3   1,8   2,7   3,6   4,5   9
%o A337765 (Ruby)
%o A337765 def A(n)
%o A337765   f_ary = [[n]]
%o A337765   cnt = 1
%o A337765   while f_ary.size > 0
%o A337765     b_ary = []
%o A337765     f_ary.each{|i|
%o A337765       s = i.size
%o A337765       (1..i[0] - 1).each{|j|
%o A337765         a = [j]
%o A337765         (0..s - 1).each{|k|
%o A337765           num = i[k] - a[k]
%o A337765           if num > 0
%o A337765             a << num
%o A337765           else
%o A337765             break
%o A337765           end
%o A337765         }
%o A337765         b_ary << a if a.size == s + 1 && a == a.sort
%o A337765       }
%o A337765     }
%o A337765     f_ary = b_ary
%o A337765     cnt += f_ary.size
%o A337765   end
%o A337765   cnt
%o A337765 end
%o A337765 def A337765(n)
%o A337765   (1..n).map{|i| A(i)}
%o A337765 end
%o A337765 p A337765(50)
%Y A337765 Cf. A062684, A062896, A337766.
%K A337765 nonn
%O A337765 1,2
%A A337765 _Seiichi Manyama_, Sep 19 2020