A047732 First differences are A005563.

1, 4, 12, 27, 51, 86, 134, 197, 277, 376, 496, 639, 807, 1002, 1226, 1481, 1769, 2092, 2452, 2851, 3291, 3774, 4302, 4877, 5501, 6176, 6904, 7687, 8527, 9426, 10386, 11409, 12497, 13652, 14876, 16171, 17539, 18982, 20502, 22101, 23781, 25544, 27392, 29327

Offset: 0

Author

Patternfinder(AT)webtv.net (Robert Newstedt)

Keywords

• nonn
• easy

Comments

Number of 3-permutations of n elements avoiding the patterns 132, 321. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
• Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000330, A005563, A051925.

Programs

Magma

I:=[1, 4, 12, 27]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012

Mathematica

CoefficientList[Series[(1+2*x^2-x^3)/((1-x)^4),{x,0,50}],x] (* Vincenzo Librandi, Jun 28 2012 *)
LinearRecurrence[{4,-6,4,-1},{1,4,12,27},50] (* Harvey P. Dale, Aug 22 2015 *)

Formula

a(n) = A051925(n+1) + 1. - Alex Ratushnyak, Jun 27 2012

From Vincenzo Librandi, Jun 28 2012: (Start)

G.f.: (1 + 2*x^2 - x^3)/(1-x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = (2*n^3 + 9*n^2 + 7*n + 6)/6. (End)

a(n) = A000330(n+1) - n. - John Tyler Rascoe, Jun 24 2022