

A026035


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


6



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,1


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(n1)*p(n). This sequence represents the minimum possible S.  Leroy Quet and Rainer Rosenthal, Jan 30 2005
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



FORMULA

a(n) = (2*n^3 + 4*n  3 + 3*(1)^n)/12.  Ralf Stephan, Jan 30 2005.
a(2)=2, a(3)=5, a(4)=12, a(5)=22, a(6)=38; for n>6, a(n) = 3*a(n1)  2*a(n2)  2*a(n3) + 3*a(n4)  a(n5).  Harvey P. Dale, May 31 2013


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 *)


PROG

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


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



