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A374438
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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.
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2
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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, 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, 42, 45, 40, 30, 10, 3, 1, 2, 3, 18, 24, 56, 63, 70, 60, 30, 15, 2
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OFFSET
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0,3
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COMMENTS
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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.
As m -> oo, the rows of the triangles become the initial segments of the integers.
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LINKS
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EXAMPLE
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Triangle starts:
[ 0] [1]
[ 1] [1, 2]
[ 2] [1, 2, 3]
[ 3] [1, 2, 3, 2]
[ 4] [1, 2, 3, 4, 3]
[ 5] [1, 2, 3, 6, 6, 2]
[ 6] [1, 2, 3, 8, 9, 6, 3]
[ 7] [1, 2, 3, 10, 12, 12, 9, 2]
[ 8] [1, 2, 3, 12, 15, 20, 18, 8, 3]
[ 9] [1, 2, 3, 14, 18, 30, 30, 20, 12, 2]
[10] [1, 2, 3, 16, 21, 42, 45, 40, 30, 10, 3]
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MAPLE
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M := 3; # family index
T := proc(n, k) option remember; if k > n then 0 elif k < M then k + 1 else
T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..11);
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PROG
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(Python)
from functools import cache
@cache
def T(n: int, k: int) -> int:
if k > n: return 0
if k < 3: return k + 1
return T(n - 1, k) + T(n - 2, k - 2)
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CROSSREFS
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Family of triangles: A162515 (m=1, Fibonacci), A374439 (m=2, Lucas), this triangle (m=3).
Row sums: A187890 (apart from initial terms), also A001060 + 1 (with 1 prepended).
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KEYWORD
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AUTHOR
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STATUS
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approved
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