cp's OEIS Frontend

A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.

%I A111284 #43 Feb 02 2023 17:14:45
%S A111284 1,2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,82,86,90,
%T A111284 94,98,102,106,110,114,118,122,126,130,134,138,142,146,150,154,158,
%U A111284 162,166,170,174,178,182,186,190,194,198,202,206,210,214,218,222,226,230
%N A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.
%C A111284 This sequence might also be called "The Non-Pythagorean integers" since no primitive Pythagorean triangle (PPT) exists containing them. Numbers of the form 4n-2 cannot be a leg or hypotenuse of PPT [a,b,c]. This excludes all even members of the present sequence. Integers 1 and zero are excluded because they form a 'degenerate triangle' with angles = 0. Compare A125667. - _H. Lee Price_, Feb 02 2007
%C A111284 Besides the first term this sequence is the denominator of Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - _Mohammad K. Azarian_, Oct 14 2011
%D A111284 Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
%D A111284 Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).
%H A111284 Vincenzo Librandi, <a href="/A111284/b111284.txt">Table of n, a(n) for n = 1..10000</a>
%H A111284 M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r31">Sorting Classes</a>, Elec. J. of Comb. 12 (2005) R31
%H A111284 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A111284 a(n) = 4*n-6, n>=2.
%F A111284 a(n) = A016825(n-2), n>1. - _R. J. Mathar_, Aug 18 2008
%F A111284 G.f.: x(1+3x^2)/(1-x)^2. - _R. J. Mathar_, Nov 10 2008
%F A111284 a(n^2 - 2n + 3)/2 = Sum_{i=1..n} a(i). - _Ivan N. Ianakiev_, Apr 24 2013
%F A111284 a(n) = 2*a(n-1) - a(n-2), n>3. - _Rick L. Shepherd_, Jul 06 2017
%F A111284 a(n) = |A161718(n-1)| = (-1)^(n-1)*A161718(n-1), n>0. - _Rick L. Shepherd_, Jul 06 2017
%F A111284 E.g.f.: 3*(x + 2) + exp(x)*(4*x - 6). - _Stefano Spezia_, Feb 02 2023
%t A111284 Table[If[n == 1, 1, 4n - 6], {n, 60}] (* _Robert G. Wilson v_, Nov 04 2005 *)
%Y A111284 Cf. A125667. Complement of the union of {1}, A020882, A020883 and A020884.
%Y A111284 Cf. A016825, A161718.
%K A111284 nonn,easy
%O A111284 1,2
%A A111284 _Len Smiley_, Nov 01 2005