This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A182617 #33 Jan 12 2025 10:21:29 %S A182617 0,5,9,12,15,18,21,23,26,29,31,34,36,39,41,44,46,49,51,53,56,58,61,63, %T A182617 65,68,70,72,75,77,79,82,84,86,89,91,93,95,98,100 %N A182617 Number of toothpicks in a toothpick spiral around n cells on hexagonal net. %C A182617 The toothpick spiral contains n hexagonal "ON" cells that are connected without holes. A hexagonal cell is "ON" if the hexagon has 6 vertices that are covered by the toothpicks. %C A182617 Attempt of an explanation: in the hexagonal grid, we can pick any of the hexagons as a center, and then define a ring of 6 first neighbors (hexagons adjacent to the center), then define a ring of 12 second neighbors (hexagons adjacent to any of the first ring) and so on. The current sequence describes a self-avoiding walk which starts in a spiral around the center hexagon, which covers 5 edges. The walk then takes one step to reach the rim of the first ring and travels once around this ring until it reaches a point where self-avoidance stops it. It then takes one step to reach the rim of the second ring and walks around that one, etc. Imagine that on each edge we place a toothpick if it's on the path, and interrupt counting the total number of toothpicks each time one of the hexagons has six vertices covered. The total number of toothpicks after n-th stage define this sequence. Note that, except from the last sentence, this comment is a copy from _R. J. Mathar_'s comment in A182618 (Dec 13 2010). - _Omar E. Pol_, Sep 15 2013 %H A182617 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.] %H A182617 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a> %H A182617 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %F A182617 Conjecture: a(n) = 2*n + ceiling(sqrt(12*n - 3)), for n > 0. - _Vincenzo Librandi_, Sep 20 2017 %e A182617 On the infinite hexagonal grid we start at stage 0 with no toothpicks, so a(0) = 0. %e A182617 At stage 1 we place 5 toothpicks on the edges of the first hexagonal cell, so a(1) = 5. %e A182617 At stage 2, from the last exposed endpoint, we place 4 other toothpicks on the edges of the second hexagonal cell, so a(2) = 5 + 4 = 9 because there are 9 toothpicks in the structure. %e A182617 At stage 3, from the last exposed endpoint, we place 3 other toothpicks on the edges of the third hexagonal cell, so a(3) = 9 + 3 = 12 because there are 12 toothpicks in the spiral. %e A182617 From _Omar E. Pol_, Sep 14 2013: (Start) %e A182617 Illustration of initial terms: %e A182617 . _ _ _ _ %e A182617 . _ _/ \ _/ \_ _/ \_ _/ \_ %e A182617 . _ _ / _ / _ / _ \ / _ \ / _ \ %e A182617 . / \ / \ \ / \ \ / \ \ / \ / \ / \ / \ / \ / %e A182617 . _/ / _/ / _/ / _/ / _/ / _/ \ / _/ \ %e A182617 . \_/ \_/ \_/ \_/ \_/ _/ \_/ _/ %e A182617 . _/ %e A182617 . %e A182617 . 5 9 12 15 18 21 23 %e A182617 . %e A182617 (End) %Y A182617 Cf. A139250, A182618, A182619, A182632, A182840. %K A182617 nonn,more %O A182617 0,2 %A A182617 _Omar E. Pol_, Dec 13 2010