

A164001


Spiral of triangles around a hexagon.


16



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
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OFFSET

1,2


COMMENTS

a(n) is the side length of the nth 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.
a(n) is the number of bitstrings of length n1 without two consecutive 0's or three consecutive 1's.  Zachary Stier, Mar 16 2021


LINKS

I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. NeumannChun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIPStern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015. See Conjecture 5.8.


FORMULA

If n < 5 then a(n) = n, otherwise a(n) = a(n2) + a(n3).
a(n) ~ 1.67873... * 1.32471...^(n1) 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


MATHEMATICA

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


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



