A164001 Spiral of triangles around a hexagon.

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426

Offset: 1

Author

Omar E. Pol, Oct 27 2009

Keywords

• easy
• nonn

Comments

a(n) is the side length of the n-th triangle in a spiral around a hexagon with side length = 1.

Sequence very similar to A134816, but without repeated terms. Records in A134816. Also records in A000931, the Padovan sequence.

Column k=2 of A242464 (with offset 0). - Alois P. Heinz, May 19 2014

a(n) is the number of bitstrings of length n-1 without two consecutive 0's or three consecutive 1's. - Zachary Stier, Mar 16 2021

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015. See Conjecture 5.8.
• Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
• Index entries for linear recurrences with constant coefficients, signature (0,1,1).

Crossrefs

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A060006.

Programs

Mathematica

LinearRecurrence[{0,1,1},{1,2,3,4},50] (* Harvey P. Dale, Jul 08 2017 *)

Formula

If n < 5 then a(n) = n, otherwise a(n) = a(n-2) + a(n-3).

G.f.: -x - 1 + (-x^2 - 2*x - 1)/(-1 + x^2 + x^3). a(n) = A000931(n+4) + A000931(n+5) = A000931(n+7), n > 1. - R. J. Mathar, Oct 29 2009

a(n) ~ 1.67873... * 1.32471...^(n-1) where 1.32471... is the real root of x^3 - x - 1 = 0 (see A060006), and 1.67873... is the real root of 23*x^3 - 46*x^2 + 13*x - 1 = 0. - Ricardo Bittencourt, May 14 2023