A047732 - OEIS

%I #39 Aug 20 2022 13:39:40

%S 1,4,12,27,51,86,134,197,277,376,496,639,807,1002,1226,1481,1769,2092,

%T 2452,2851,3291,3774,4302,4877,5501,6176,6904,7687,8527,9426,10386,

%U 11409,12497,13652,14876,16171,17539,18982,20502,22101,23781,25544,27392,29327

%N First differences are A005563.

%C Number of 3-permutations of n elements avoiding the patterns 132, 321. See Bonichon and Sun. - _Michel Marcus_, Aug 19 2022

%H Vincenzo Librandi, <a href="/A047732/b047732.txt">Table of n, a(n) for n = 0..1000</a>

%H Nicolas Bonichon and Pierre-Jean Morel, <a href="https://arxiv.org/abs/2202.12677">Baxter d-permutations and other pattern avoiding classes</a>, arXiv:2202.12677 [math.CO], 2022.

%H Nathan Sun, <a href="https://arxiv.org/abs/2208.08506">On d-permutations and Pattern Avoidance Classes</a>, arXiv:2208.08506 [math.CO], 2022.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = A051925(n+1) + 1. - _Alex Ratushnyak_, Jun 27 2012

%F From _Vincenzo Librandi_, Jun 28 2012: (Start)

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

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

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

%F a(n) = A000330(n+1) - n. - _John Tyler Rascoe_, Jun 24 2022

%t CoefficientList[Series[(1+2*x^2-x^3)/((1-x)^4),{x,0,50}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)

%t LinearRecurrence[{4,-6,4,-1},{1,4,12,27},50] (* _Harvey P. Dale_, Aug 22 2015 *)

%o (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

%Y Cf. A000330, A005563, A051925.

%K nonn,easy

%O 0,2

%A Patternfinder(AT)webtv.net (Robert Newstedt)