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A164991
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Number of triangular involutions of n. A triangular involution is a square involution with at most three faces.
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2
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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
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OFFSET
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1,3
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COMMENTS
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The sequence 2^(n+1) - binomial(n, floor(n/2)), which begins 1,3,6,... has Hankel transform (-1)^n*(2*n+1) (A157142). - Paul Barry, Nov 03 2010
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n) = 2^(n-1) - binomial(n-2, floor((n-2)/2)) for n>1, a(1)=1.
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)
(-n+1)*a(n) + 2*(n-1)*a(n-1) + 4*(n-4)*a(n-2) + 8*(-n+4)*a(n-3) = 0. - R. J. Mathar, Aug 09 2017
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MATHEMATICA
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Join[{1}, Table[2^(n-1)-Binomial[n-2, Floor[(n-2)/2]], {n, 2, 30}]] (* Harvey P. Dale, Dec 26 2015 *)
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PROG
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(PARI) a(n) = 2^(n-1) - binomial(n-2, (n-2)\2) \\ Michel Marcus, May 27 2013
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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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Simone Rinaldi (rinaldi(AT)unisi.it), Sep 04 2009
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STATUS
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approved
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