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A028399
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a(n) = 2^n - 4.
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31
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0, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068, 262140, 524284, 1048572, 2097148, 4194300, 8388604, 16777212, 33554428, 67108860, 134217724, 268435452, 536870908, 1073741820, 2147483644, 4294967292, 8589934588, 17179869180
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OFFSET
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2,2
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COMMENTS
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Number of permutations of [n] with 2 sequences.
Number of 2 X n binary matrices that avoid simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
The number of edges in the dual Edwards-Venn diagram graph with n-1 digits when n>2.
a(n) (n>=6) is the number of vertices in the molecular graph NS2[n-5], defined pictorially in the Ashrafi et al. reference (Fig. 2, where NS2[2] is shown). - Emeric Deutsch, May 16 2018
With regard to the comment above about a(n) being the "number of permutations of [n] with 2 sequences", we refer to Ex. 13 (pp. 260-261) of Comtet (1974), who uses the definition of a "séquence" given by André in several of his papers in the 19th century.
In the terminology of array A059427, these so-called "séquences" in permutations (defined by Comtet and André) are called "alternating runs" (or just "runs"). We discuss these so-called "séquences" below.
If b = (b_1, b_2, ..., b_n) is a permutation of [n], written in one-line notation (not in cycle notation), a "séquence" in a permutation of length l >= 2 (according to Comtet) is a maximal interval of integers {i, i+1, ..., i+l-1} for some i (where 1 <= i <= n-l+1) such that b_i < b_{i+1} < ... < b_{i+l-1} or b_i > b_{i+1} > ... > b_{i+l-1}. (The word "maximal" means that, in the first case, we have b_{i-1} > b_i and b_{i+l} < b_{i+l-1}, while in the second case, we have b_{i-1} < b_i and b_{i+l} > b_{i+l-1} provided b_{i-1} and b_{i+l} can be defined.)
When defining a "séquence", André (1884) actually refers to the list of terms (b_i, b_{i+1}, ..., b_{i+l-1}) rather than the corresponding index set {i, i+1, ..., i+l-1} (which is essentially the same thing).
For more details about these so-called "séquences" (or "alternate runs"), see the comments and examples for sequence A000708.
(End)
For n >= 1, a(n+2) is the number of shortest paths from (0,0) of a square grid to {(x,y): |x|+|y| = n} along the grid line. - Jianing Song, Aug 23 2021
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261.
A. W. F. Edwards, Cogwheels of the Mind, Johns Hopkins University Press, 2004, p. 82.
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LINKS
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FORMULA
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EXAMPLE
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We have a(3) = 4 because each of the following permutations of [3] has the following so-called "séquences" ("alternate runs"):
123 -> 123 (one),
132 -> 13, 32 (two),
213 -> 21, 13 (two),
231 -> 23, 31 (two),
312 -> 31, 12 (two),
321 -> 321 (one).
Recall that a so-called "séquence" ("alternate run") must start with a "maximum" and end with "minimum", or vice versa, and it should not contain any other maxima and minima in between. Two consecutive such "séquences" ("alternate runs") have exactly one minimum or exactly one maximum in common.
(End)
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MAPLE
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MATHEMATICA
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PROG
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(PARI) a(n)=if(n<2, 0, 2^n-4)
(Python)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Additional comments from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 02 2001
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STATUS
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approved
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