# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a360021 Showing 1-1 of 1 %I A360021 #17 Feb 11 2023 20:31:58 %S A360021 1,6,45,315,2205,15624,111888,807840,5868720,42799680,312504192, %T A360021 2278418688,16549827840,119567831040,858293084160,6118081708032, %U A360021 43298650386432,304260332175360,2123395686236160,14722247331348480,101446590051975168,695007859780878336,4735844958575001600 %N A360021 Number of unordered triples of self-avoiding paths with nodes that cover all vertices of a convex n-gon; one-node paths are allowed. %H A360021 Ivaylo Kortezov, Sets of Non-self-intersecting Paths Connecting the Vertices of a Convex Polygon, Mathematics and Informatics, Vol. 65, No. 6, 2022. %H A360021 Index entries for linear recurrences with constant coefficients, signature (48,-1040,13440,-115296,691200,-2967296,9185280,-20336896,31395840,-32071680,19464192,-5308416). %F A360021 a(n) = n*(n-1)*(n-2)*2^(n-10)*(3^(n-4) + 3*2^(n-3) + 9) for n >= 4. %e A360021 a(5) = 5!/(2!2!2!) + binomial(5,2)*3 = 15 + 30 = 45; the first summand corresponds to the case when two of the paths have two nodes each and one path has one node; the second corresponds to the case when two of the paths have one node each and one path has three nodes. %t A360021 LinearRecurrence[{48,-1040,13440,-115296,691200,-2967296,9185280,-20336896,31395840,-32071680,19464192,-5308416},{1,6,45,315,2205,15624,111888,807840,5868720,42799680,312504192,2278418688,16549827840},23] (* _Stefano Spezia_, Jan 22 2023 *) %o A360021 (PARI) a(n) = if(n==3, 1, n*(n-1)*(n-2)*2^(n-10)*(3^(n-4) + 3*2^(n-3) + 9)) \\ _Andrew Howroyd_, Jan 23 2023 %Y A360021 Cf. A359405 (unordered pairs). %K A360021 nonn,easy %O A360021 3,2 %A A360021 _Ivaylo Kortezov_, Jan 22 2023 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE