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.

1, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset

1,2

Comments

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

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

References

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.

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 4*n-6, n>=2.

a(n) = A016825(n-2), n>1. - R. J. Mathar, Aug 18 2008

G.f.: x(1+3x^2)/(1-x)^2. - R. J. Mathar, Nov 10 2008

a(n^2 - 2n + 3)/2 = Sum_{i=1..n} a(i). - Ivan N. Ianakiev, Apr 24 2013

a(n) = 2*a(n-1) - a(n-2), n>3. - Rick L. Shepherd, Jul 06 2017

a(n) = |A161718(n-1)| = (-1)^(n-1)*A161718(n-1), n>0. - Rick L. Shepherd, Jul 06 2017

E.g.f.: 3*(x + 2) + exp(x)*(4*x - 6). - Stefano Spezia, Feb 02 2023

Mathematica

Table[If[n == 1, 1, 4n - 6], {n, 60}] (* Robert G. Wilson v, Nov 04 2005 *)

Crossrefs

Cf. A125667. Complement of the union of {1}, A020882, A020883 and A020884.

Cf. A016825, A161718.

Sequence in context: A187884 A068977 A251538 * A130824 A016825 A161718

Adjacent sequences: A111281 A111282 A111283 * A111285 A111286 A111287

Keyword

nonn,easy

Author

Len Smiley, Nov 01 2005

Status

approved