2,1

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.

Comments from Dmitry Kamenetsky (Dmitry.Kamenetsky(AT)rsise.anu.edu.au), 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)

(1/12) [2n^3 + 4n - 3 + 3(-1)^n ]. - Ralf Stephan, Jan 30 2005

f(n) = (2n^3 + 4n - 3 + 3(-1)^n)/12. - Ralf Stephan, Jan 31 2005.

CoefficientList[ Series[(2 - x + x^2)/((1 + x)(1 - x)^4), {x, 0, 45}], x] (from Robert G. Wilson v Jan 29 2005)

Cf. A101986.

Sequence in context: A116729 A048840 A116718 this_sequence A086734 A123647 A116711

Adjacent sequences: A026032 A026033 A026034 this_sequence A026036 A026037 A026038

nonn

Clark Kimberling (ck6(AT)evansville.edu)

Corrected by Ralf Stephan, Jan 09 2005