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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(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, M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, t_{N0}(n,5) in theorem 5.
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(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - Harvey P. Dale, May 31 2013
a(n) = binomial(n,3) + floor(n^2/2). - 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 *)
PROG
(Magma) [Binomial(n, 3)+Floor(n^2/2): n in [2..50]]; // Bruno Berselli, Jun 08 2017
CROSSREFS
Cf. A101986.
Sequence in context: A116729 A048840 A116718 * A215183 A086734 A123647
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Corrected by Ralf Stephan, Jan 09 2005
STATUS
approved

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