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A181510
Number of permutations of the multiset {1,1,2,2,3,3,...,n+1,n+1} avoiding the permutation patterns {132, 231, 2134}.
7
6, 18, 34, 54, 78, 106, 138, 174, 214, 258, 306, 358, 414, 474, 538, 606, 678, 754, 834, 918, 1006, 1098, 1194, 1294, 1398, 1506, 1618, 1734, 1854, 1978, 2106, 2238, 2374, 2514, 2658, 2806, 2958, 3114, 3274, 3438, 3606, 3778, 3954, 4134, 4318, 4506, 4698, 4894
OFFSET
1,1
COMMENTS
a(n) is also the surface ares of the n-th solid in the following recursive construction:
The first solid is a unit cube (hence a(1)=6).
To form the n-th solid from the (n-1)st solid, construct a row of 2n-1 cubes, then center the (n-1)st solid on top of this row. (For example, the second solid is a row of 3 unit cubes, with a single unit cube centered on top of the middle cube. This construction has surface area a(2)=18.)
The sequence provides all nonnegative integers m such that 2*m+13 is a square. - Bruno Berselli, Mar 01 2013
LINKS
FORMULA
a(n) = 2n^2+6n-2.
From Bruno Berselli, Oct 29 2010: (Start)
G.f.: 2*x*(3-x^2)/(1-x)^3.
a(n) -3*a(n-1) +3*a(n-2) -a(n-3) = 0 for n>3.
a(n) = 2*A014209(n) = 2*A082111(n-1)+4 = A051936(2n+2)+n+4. (End)
Sum_{n>=1} 1/a(n) = 2/3 + Pi*tan(sqrt(13)*Pi/2)/(2*sqrt(13)). - Amiram Eldar, Dec 23 2022
EXAMPLE
For n=1, the permutations of {1,1,2,2} avoiding the patterns {132, 231, 2134} are {1122, 1212, 1221, 2112, 2121, 2211}.
For n=2, the permutations of {1,1,2,2,3,3} avoiding the patterns {132, 231, 2134} are {112233, 121233, 122133, 211233, 212133, 221133, 311223, 312123, 312213, 321123, 321213, 322113, 331122, 331212, 331221, 332112, 332121, 332211}.
MATHEMATICA
a[n_] := 2*n^2 + 6*n - 2; Array[a, 60] (* Amiram Eldar, Dec 23 2022 *)
LinearRecurrence[{3, -3, 1}, {6, 18, 34}, 50] (* Harvey P. Dale, May 10 2023 *)
PROG
(PARI) a(n)=2*n^2+6*n-2 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lara Pudwell, Oct 25 2010
STATUS
approved