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A026519
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Irregular triangular array T read by rows: T(i,0) = T(i,2i)=1 for i >= 0; T(i,1) = T(i,2i-1) = floor[(i+1)/2] for i >= 1; for even n >= 2, T(i,j) = T(i-1,j-2) + T(i-1,j) for j = 2..2i-2; for odd n >= 3, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) for j = 2..2i-2.
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26
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 4, 14, 32, 65, 104, 150, 180, 196, 180
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OFFSET
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1,7
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COMMENTS
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T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i)-s(i-1)| = 1 if i is even, |s(i)-s(i-1)| <= 1 if i is odd.
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REFERENCES
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Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
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LINKS
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Clark Kimberling, Rows 0..100, flattened Index entries for triangles and arrays related to Pascal's triangle
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EXAMPLE
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First 5 rows:
1
1 ... 1 ... 1
1 ... 1 ... 2 ... 1 ... 1
1 ... 2 ... 4 ... 4 ... 4 ... 2 ... 1
1 ... 2 ... 5 ... 6 ... 8 ... 6 ... 5 ... 2 ... 1
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MATHEMATICA
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z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[(n + 1)/2]; t[n_, k_] := Floor[(n + 1)/2] /; k == 2 n - 1; t[n_, k_] := t[n, k] = If[EvenQ[n], t[n - 1, k - 2] + t[n - 1, k], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u] (* A026519 array *)
v = Flatten[u] (* A026519 sequence *)
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CROSSREFS
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Cf. A026527, A026552, A026536, A026568, A026584, A027926.
Sequence in context: A267383 A272896 A188919 * A025177 A026148 A117211
Adjacent sequences: A026516 A026517 A026518 * A026520 A026521 A026522
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KEYWORD
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nonn,tabf
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Updated by Clark Kimberling, Aug 29 2014
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STATUS
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approved
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