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 A164001 #52 Jan 10 2025 04:38:58
%S A164001 1,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,616,
%T A164001 816,1081,1432,1897,2513,3329,4410,5842,7739,10252,13581,17991,23833,
%U A164001 31572,41824,55405,73396,97229,128801,170625,226030,299426
%N A164001 Spiral of triangles around a hexagon.
%C A164001 a(n) is the side length of the n-th triangle in a spiral around a hexagon with side length = 1.
%C A164001 Sequence very similar to A134816, but without repeated terms. Records in A134816. Also records in A000931, the Padovan sequence.
%C A164001 Column k=2 of A242464 (with offset 0). - _Alois P. Heinz_, May 19 2014
%C A164001 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
%H A164001 Alois P. Heinz, <a href="/A164001/b164001.txt">Table of n, a(n) for n = 1..1000</a>
%H A164001 I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, <a href="http://arxiv.org/abs/1509.05239">Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences</a>, arXiv:1509.05239v1 [math.CO] 17 Sep 2015. See Conjecture 5.8.
%H A164001 Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, <a href="https://doi.org/10.9734/ajpas/2024/v26i12691">A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers</a>, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
%H A164001 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1).
%F A164001 If n < 5 then a(n) = n, otherwise a(n) = a(n-2) + a(n-3).
%F A164001 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
%F A164001 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
%t A164001 LinearRecurrence[{0,1,1},{1,2,3,4},50] (* _Harvey P. Dale_, Jul 08 2017 *)
%Y A164001 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.
%Y A164001 Cf. A060006.
%K A164001 easy,nonn
%O A164001 1,2
%A A164001 _Omar E. Pol_, Oct 27 2009