A026035 Expansion of x^2*(2 - x + x^2) / ((1 + x)*(1 - x)^4).

2, 5, 12, 22, 38, 59, 88, 124, 170, 225, 292, 370, 462, 567, 688, 824, 978, 1149, 1340, 1550, 1782, 2035, 2312, 2612, 2938, 3289, 3668, 4074, 4510, 4975, 5472, 6000, 6562, 7157, 7788, 8454, 9158, 9899, 10680, 11500, 12362, 13265, 14212, 15202, 16238

Offset: 2

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Equals (d(n)-r(n))/2, where d = A006527 and r is the periodic sequence with fundamental period (0,1,0,1).

Consider any of the permutations of (1,2,3,...,n) as p(1),p(2),p(3),...,p(n). Then take the sum S of products formed from the permutation as S = p(1)*p(2) + p(2)*p(3) + p(3)*p(4) +... + p(n-1)*p(n). This sequence represents the minimum possible S. - Leroy Quet and Rainer Rosenthal, Jan 30 2005

From Dmitry Kamenetsky, Dec 15 2006: (Start)

This sequence is related to A101986, except here we take the minimum sum of products of successive pairs. Here is a method for generating such permutations. Start with two lists, the first has numbers 1 to n, while the second is empty.

Repeat the following operations until the first list is empty:

1. Move the largest number of the first list to the leftmost available position in the second list. The move operation removes the original number from the first list.

2. Move the largest number of the first list to the rightmost available position in the second list.

3. Move the smallest number of the first list to the leftmost available position in the second list.

4. Move the smallest number of the first list to the rightmost available position in the second list. For example when n=8, the permutation is 8, 1, 6, 3, 4, 5, 2, 7.

(End)

Links

• M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, t_{N0}(n,5) in theorem 5.
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A101986.

Programs

Magma

[Binomial(n,3)+Floor(n^2/2): n in [2..50]]; // Bruno Berselli, Jun 08 2017

Mathematica

CoefficientList[Series[(2 - x + x^2)/((1 + x) (1 - x)^4), {x, 0, 45}], x] (* Robert G. Wilson v, Jan 29 2005 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {2, 5, 12, 22, 38}, 50] (* Harvey P. Dale, May 31 2013 *)
Table[(2 n^3 + 4 n - 3 + 3 (-1)^n)/12, {n, 2, 50}] (* Bruno Berselli, Jun 08 2017 *)

Formula

a(n) = (2*n^3 + 4*n - 3 + 3*(-1)^n)/12. - Ralf Stephan, Jan 30 2005.

For n>6, a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5), and a(2)=2, a(3)=5, a(4)=12, a(5)=22, a(6)=38. - Harvey P. Dale, May 31 2013

a(n) = binomial(n,3) + floor(n^2/2). - Bruno Berselli, Jun 08 2017

Extensions

Corrected by Ralf Stephan, Jan 09 2005