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A026519 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. 26
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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.

REFERENCES

Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

LINKS

Clark Kimberling, Rows 0..100, flattened Index entries for triangles and arrays related to Pascal's triangle

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A026527, A026552, A026536, A026568, A026584, A027926.

Sequence in context: A267383 A272896 A188919 * A025177 A026148 A117211

Adjacent sequences:  A026516 A026517 A026518 * A026520 A026521 A026522

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling

EXTENSIONS

Updated by Clark Kimberling, Aug 29 2014

STATUS

approved

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Last modified September 22 03:57 EDT 2017. Contains 292328 sequences.