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A164991 Number of triangular involutions of n. A triangular involution is a square involution with at most three faces. 1
1, 1, 3, 6, 13, 26, 54, 108, 221, 442, 898, 1796, 3634, 7268, 14668, 29336, 59101, 118202, 237834, 475668, 956198, 1912396, 3841588, 7683176, 15425138, 30850276, 61908564, 123817128, 248377156, 496754312 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The sequence 2^(n+1)-C(n,floor(n/2)), which begins 1,3,6,... has Hankel transform (-1)^n*(2n+1) (A157142). - Paul Barry, Nov 03 2010

For n >= 2 also row sums of A258445. - Wolfdieter Lang, Jun 27 2015

REFERENCES

F. Disanto, A. Frosini, S. Rinaldi, Square Involutions, Proceedings of Permutation Patterns, July, 13-17 2009, Florence.

T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, 19th Conference on Formal Power Series and Algebraic Combinatorics, Tianjin, China, July 2-6, 2007.

LINKS

Table of n, a(n) for n=1..30.

T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, arXiv:math/0603225 [math.CO]

FORMULA

a(n) = 2^(n-1) - C(n-2,(n-2)/2).

From Wolfdieter Lang, Jun 27 2015: (Start)

a(n) = Sum_{k = 1..2*n-3} A258445(n-1, k), n >= 2.

a(2*k+1) = 4*Sum{j = 0..(k-2)} binomial(2*k-1,j) + 3*binomial(2*k-1,k-1), k >= 1.

a(2*k) = 4*Sum{j = 0..(k-2)} binomial(2*(k-1),j) + binomial(2*(k-1),k-1), k >= 1. (End)

MATHEMATICA

Join[{1}, Table[2^(n-1)-Binomial[n-2, Floor[(n-2)/2]], {n, 2, 30}]] (* Harvey P. Dale, Dec 26 2015 *)

PROG

(PARI) a(n) =  2^(n-1) - binomial(n-2, (n-2)\2) \\ Michel Marcus, May 27 2013

CROSSREFS

Cf. A128652, A128650, A258445.

Sequence in context: A081254 A125049 A267581 * A213255 A215985 A215986

Adjacent sequences:  A164988 A164989 A164990 * A164992 A164993 A164994

KEYWORD

easy,nonn

AUTHOR

Simone Rinaldi (rinaldi(AT)unisi.it), Sep 04 2009

STATUS

approved

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Last modified March 27 12:14 EDT 2017. Contains 284176 sequences.