# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a284843 Showing 1-1 of 1 %I A284843 #26 May 11 2021 08:56:31 %S A284843 3,12,55,318,2163,16952,150183,1483290,16153027,192234372,2481692343, %T A284843 34538108822,515428409715,8210201868528,139028223047143, %U A284843 2493848817565362,47236883312958723,942120664413581180,19735072617803491767,433187081497364626542 %N A284843 Number of permutations on [n+2] with no circular 2-successions. %C A284843 Define a circular k-succession in a permutation p on [n] as either a pair p(i), p(i+1) if p(i+1) = p(i) + k, or as the pair p(n), p(1) if p(1) = p(n) + k. If we let d*(n,k) be the number of permutations on [n] that avoid substrings (j, j+k), 1 <= j <= n, k = 2, i.e., permutations with no circular 2-succession, then a(n) counts d*(n+2, 2). %C A284843 For example, for n = 1, the permutations in S3 that contain the substring {13} in circular 2-succession are 132, 213, 321, therefore d*(3,2) consists of the permutations 123, 231, 312, and a(1) = 3. %H A284843 Enrique Navarrete, Generalized K-Shift Forbidden Substrings in Permutations, arXiv:1610.06217 [math.CO], 2016. %F A284843 a(n) = (n+2) * Sum_{j = 0..n} (-1)^j * (n - j + 1) * n!/j!. %e A284843 For n = 2, the permutations in S4 that contain the substrings {13, 24} in circular 2-successions are 1243, 1324, 1342, 2134, 2413, 2431, 3124, 3241, 3421, 4132, 4213, 4312, therefore d*(4,2) consists of the complementary permutations in S4, and a(2) = 12. %t A284843 Table[(n + 2) Sum[(-1)^j * (n - j + 1) * n!/j!, {j, 0, n}], {n, 20}] (* _Michael De Vlieger_, Apr 05 2017 *) %o A284843 (PARI) a(n) = (n + 2) * sum(j=0, n, (-1)^j * (n - j + 1) * n!/j!); \\ _Indranil Ghosh_, Apr 06 2017 %o A284843 (Python) %o A284843 from sympy import factorial %o A284843 print([(n + 2) * sum([(-1)**j * (n - j + 1) * factorial(n)//factorial(j) for j in range(n + 1)]) for n in range(1, 50)]) # _Indranil Ghosh_, Apr 06 2017 %K A284843 nonn %O A284843 1,1 %A A284843 _Enrique Navarrete_, Apr 03 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE