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%I #14 Jul 22 2024 08:18:56
%S 1,1,2,1,2,3,1,2,3,2,1,2,3,4,3,1,2,3,6,6,2,1,2,3,8,9,6,3,1,2,3,10,12,
%T 12,9,2,1,2,3,12,15,20,18,8,3,1,2,3,14,18,30,30,20,12,2,1,2,3,16,21,
%U 42,45,40,30,10,3,1,2,3,18,24,56,63,70,60,30,15,2
%N Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with initial values T(n, k) = k + 1 for k < 3.
%C See A374439 and the cross-references for comments about this family of triangles, where the recurrence is defined as in the name, but with an additional parameter m for the initial values: T(n, k) = k + 1 for k < m.
%C As m -> oo, the rows of the triangles become the initial segments of the integers.
%e Triangle starts:
%e [ 0] [1]
%e [ 1] [1, 2]
%e [ 2] [1, 2, 3]
%e [ 3] [1, 2, 3, 2]
%e [ 4] [1, 2, 3, 4, 3]
%e [ 5] [1, 2, 3, 6, 6, 2]
%e [ 6] [1, 2, 3, 8, 9, 6, 3]
%e [ 7] [1, 2, 3, 10, 12, 12, 9, 2]
%e [ 8] [1, 2, 3, 12, 15, 20, 18, 8, 3]
%e [ 9] [1, 2, 3, 14, 18, 30, 30, 20, 12, 2]
%e [10] [1, 2, 3, 16, 21, 42, 45, 40, 30, 10, 3]
%p M := 3; # family index
%p T := proc(n, k) option remember; if k > n then 0 elif k < M then k + 1 else
%p T(n - 1, k) + T(n - 2, k - 2) fi end:
%p seq(seq(T(n, k), k = 0..n), n = 0..11);
%o (Python)
%o from functools import cache
%o @cache
%o def T(n: int, k: int) -> int:
%o if k > n: return 0
%o if k < 3: return k + 1
%o return T(n - 1, k) + T(n - 2, k - 2)
%Y Family of triangles: A162515 (m=1, Fibonacci), A374439 (m=2, Lucas), this triangle (m=3).
%Y Row sums: A187890 (apart from initial terms), also A001060 + 1 (with 1 prepended).
%Y Cf. A006355 (odd sums), A187893 (even sums).
%Y Cf. related to deltas: A065220, A210673.
%K nonn,tabl,new
%O 0,3
%A _Peter Luschny_, Jul 22 2024
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