A141135 Minimal number of unit edges required to construct n regular pentagons when allowing edge-sharing.
5, 9, 13, 17, 21, 24, 28, 32, 36, 39, 43, 47, 50, 54, 58, 61, 65, 69, 72, 76, 80, 83, 87, 90, 94, 98, 101, 105, 109, 112
Offset: 1
Examples
a(6) = 24 since the first pentagon requires 5 edges, the 2nd, 3rd, 4th and 5th pentagons require an additional 4 edges each and the 6th pentagon requires 3 edges since it can share 2 edges (if one tiles via a 6-cycle). Thus 24 = 5 + 4 + 4 + 4 + 4 + 3.
Links
- Ralph H. Buchholz, Spiral polygon series, preprint 1985 SMJ 31, School Mathematics Journal, 1995.
- Ralph H. Buchholz and Warwick de Launey, Edge minimization, June 1996, (revised June 2008).
- Ralph H. Buchholz and Warwick de Launey, An edge minimization problem for regular polygons, The Electronic Journal of Combinatorics, Volume 16, Issue 1 (2009), #R90.
Crossrefs
Formula
Conjectures from Colin Barker, Apr 05 2019: (Start)
G.f.: x*(5 + 4*x + 4*x^2 - x^3 - x^5 + x^8 - x^9) / ((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>10.
(End)
Conjecture: if n is a term in A121149, a(n) = a(n-1) + 3, otherwise a(n) = a(n-1) + 4. - Jinyuan Wang, Apr 05 2019
Extensions
a(21)-a(30) from Jinyuan Wang, Apr 05 2019