A182617 Number of toothpicks in a toothpick spiral around n cells on hexagonal net.
0, 5, 9, 12, 15, 18, 21, 23, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 53, 56, 58, 61, 63, 65, 68, 70, 72, 75, 77, 79, 82, 84, 86, 89, 91, 93, 95, 98, 100
Offset: 0
Examples
On the infinite hexagonal grid we start at stage 0 with no toothpicks, so a(0) = 0. At stage 1 we place 5 toothpicks on the edges of the first hexagonal cell, so a(1) = 5. 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. 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. From _Omar E. Pol_, Sep 14 2013: (Start) Illustration of initial terms: . _ _ _ _ . _ _/ \ _/ \_ _/ \_ _/ \_ . _ _ / _ / _ / _ \ / _ \ / _ \ . / \ / \ \ / \ \ / \ \ / \ / \ / \ / \ / \ / . _/ / _/ / _/ / _/ / _/ / _/ \ / _/ \ . \_/ \_/ \_/ \_/ \_/ _/ \_/ _/ . _/ . . 5 9 12 15 18 21 23 . (End)
Links
- David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]
- N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
- Index entries for sequences related to toothpick sequences
Formula
Conjecture: a(n) = 2*n + ceiling(sqrt(12*n - 3)), for n > 0. - Vincenzo Librandi, Sep 20 2017
Comments