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A360173 Irregular triangle (an infinite binary tree) read by rows. The tree has root node 0, in row n=0. Each node then has left child m - n if nonnegative and right child m + n. Where m is the value of the parent node and n is the row of the children.

%I A360173 #30 Apr 16 2025 09:04:21
%S A360173 0,1,3,0,6,4,2,10,9,7,5,15,3,15,1,13,11,9,21,10,8,22,8,6,20,4,18,2,16,
%T A360173 14,28,2,18,0,16,14,30,0,16,14,12,28,12,10,26,10,8,24,6,22,20,36,11,9,
%U A360173 27,9,7,25,5,23,21,39,9,7,25,5,23,3,21,19,37,3,21
%N A360173 Irregular triangle (an infinite binary tree) read by rows. The tree has root node 0, in row n=0. Each node then has left child m - n if nonnegative and right child m + n. Where m is the value of the parent node and n is the row of the children.
%C A360173 A node will have a left child only if the value of that child is greater than or equal to 0. But, each node will have a right child, since adding n will always be greater than 0.
%C A360173 The n-th row will have A141002(n) nodes. The leftmost border is A008344 and the rightmost is A000217.
%H A360173 Rémy Sigrist, <a href="/A360173/b360173.txt">Table of n, a(n) for n = 0..9395</a> (rows for n = 0..17 flattened)
%e A360173 The binary tree starts with root 0 in row n = 0. In row n = 3, the parent node m = 3 has the first left child since 3 - 3 >= 0.
%e A360173 The tree begins:
%e A360173 row
%e A360173 [n]
%e A360173 [0]           0
%e A360173                \
%e A360173 [1]             1
%e A360173                  \
%e A360173 [2]            ___3___
%e A360173               /       \
%e A360173              /         \
%e A360173 [3]         0         __6__
%e A360173              \       /     \
%e A360173 [4]           4     2      10
%e A360173                \     \    /  \
%e A360173 [5]             9     7  5    15
%p A360173 T:= proc(n) option remember; `if`(n=0, 0, map(x->
%p A360173       [`if`(x<n, [][], x-n), x+n][], [T(n-1)])[])
%p A360173     end:
%p A360173 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jan 30 2023
%t A360173 row[n_] := Module[{r = {0}}, For[h = 1, h <= n, h++, r = Flatten[If[#-h >= 0, {#-h, #+h}, {#+h}]& /@ r]]; r];
%t A360173 Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 16 2025, after _Rémy Sigrist_ *)
%o A360173 (Python)
%o A360173 def A360173_rowlist(row_n):
%o A360173     A = [[0]]
%o A360173     for i in range(0,row_n):
%o A360173         A.append([])
%o A360173         for j in range(0,len(A[i])):
%o A360173             x = A[i][j]
%o A360173             if x - i -1 >= 0:
%o A360173                 A[i+1].append(x-i-1)
%o A360173             if x + i + 1 >= 0:
%o A360173                 A[i+1].append(x+i+1)
%o A360173     return(A)
%o A360173 (PARI) row(n) = { my (r=[0]); for (h=1, n, r=concat(apply(v->if (v-h>=0, [v-h,v+h], [v+h]), r))); return (r) } \\ _Rémy Sigrist_, Jan 31 2023
%Y A360173 Cf. A000217, A008344, A141001, A141002.
%Y A360173 Row sums give A360229.
%K A360173 nonn,look,tabf,easy
%O A360173 0,3
%A A360173 _John Tyler Rascoe_, Jan 28 2023