A337766 - OEIS

%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