A365273 Number of vertices in the Laakso graph of order n.
6, 30, 174, 1038, 6222, 37326, 223950, 1343694, 8062158, 48372942, 290237646, 1741425870, 10448555214, 62691331278, 376147987662, 2256887925966, 13541327555790, 81247965334734, 487487792008398, 2924926752050382
Offset: 1
Examples
The order 1 Laakso graph L_1 has 6 vertices and 6 edges. L_(n+1) is obtained from L_n by replacing each edge in L_n with a copy of L_1. This gives us 6 vertices, then 30, then 174, and so on.
Links
- Paolo Xausa, Table of n, a(n) for n = 1..1000
- Y. Bartal, L.-A. Gottlieb, and O. Neiman, On the Impossibility of Dimension Reduction for Doubling Subsets of l_p, SIAM Journal on Discrete Mathematics, vol. 29, no. 3. Society for Industrial & Applied Mathematics (SIAM), pp. 1207-1222, Jan. 2015.
- F. Baudier, K. Swieçicki, and A. Swift, No dimension reduction for doubling subsets of $\ell_q$ when q > 2 revisited, Journal of Mathematical Analysis and Applications, vol. 504, no. 2. Elsevier BV, p. 125407, Dec. 2021.
- S. J. Dilworth, D. Kutzarova, and M. I. Ostrovskii, Lipschitz-free Spaces on Finite Metric Spaces, Canadian Journal of Mathematics, vol. 72, no. 3. Canadian Mathematical Society, pp. 774-804, Feb. 13, 2019.
- Stephen J. Dilworth, Denka Kutzarova, and Mikhail I. Ostrovskii, Analysis on Laakso graphs with application to the structure of transportation cost spaces, arXiv:2007.07949 [math.FA], 2020-2021. See drawing of L_1 on page 4.
- S. J. Dilworth, D. Kutzarova, and M. I. Ostrovskii, Analysis on Laakso graphs with application to the structure of transportation cost spaces, Positivity 25, 1403-1435 (2021).
- S. J. Dilworth, D. Kutzarova, and S. Stankov, Metric embeddings of Laakso graphs into Banach spaces. Banach J. Math. Anal. 16, 60 (2022).
- T. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality, GAFA, Geom. funct. anal. 10, 111-123 (2000).
- U. Lang and C. Plaut, Bilipschitz Embeddings of Metric Spaces into Space Forms, Geometriae Dedicata 87, 285-307 (2001).
- A. Margaris and J. C. Robinson, Some comments on Laakso graphs and sets of differences, Comptes Rendus. Mathématique, vol. 358, no. 4. Cellule MathDoc/CEDRAM, pp. 515-521, Jul. 28, 2020.
- MathOverflow, Implementation of the "Laakso Graph" in Python, 2023.
- O. Neiman, Low Dimensional Embeddings of Doubling Metrics, Theory Comput Syst 58, 133-152 (2016).
- Th. Schlumprecht and G. Tresch, “Stochastic Embeddings of Graphs into Trees.” arXiv preprint arXiv:2306.06222 [math.CO], 2023.
- Index entries for linear recurrences with constant coefficients, signature (7,-6).
Crossrefs
Equals twice A152596.
Programs
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Mathematica
LinearRecurrence[{7,-6},{6,30},30] (* Paolo Xausa, Oct 16 2023 *)
Formula
a(n) = a(n-1) + 4*6^(n-1).
a(n) = (2/5) * (2*6^n+3). - Christian Krause, Sep 30 2023
Comments