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 A365273 #32 Oct 16 2023 06:08:51
%S A365273 6,30,174,1038,6222,37326,223950,1343694,8062158,48372942,290237646,
%T A365273 1741425870,10448555214,62691331278,376147987662,2256887925966,
%U A365273 13541327555790,81247965334734,487487792008398,2924926752050382
%N A365273 Number of vertices in the Laakso graph of order n.
%C A365273 This can be proved using the definition of the Laakso graph. The Laakso graph of level 0 is two vertices joined by an edge. The level 1 Laakso graph L_1 is obtained by replacing part of the edge of L_0 with a 4-cycle. Then the Laakso graph L_(n+1) is obtained from L_n by replacing each edge {uv} in L_n with a copy of the graph L_1, where u and v are identified with the vertices of degree 1 in L_1.
%H A365273 Paolo Xausa, <a href="/A365273/b365273.txt">Table of n, a(n) for n = 1..1000</a>
%H A365273 Y. Bartal, L.-A. Gottlieb, and O. Neiman, <a href="https://doi.org/10.1137/140977655">On the Impossibility of Dimension Reduction for Doubling Subsets of l_p</a>, SIAM Journal on Discrete Mathematics, vol. 29, no. 3. Society for Industrial & Applied Mathematics (SIAM), pp. 1207-1222, Jan. 2015.
%H A365273 F. Baudier, K. Swieçicki, and A. Swift, <a href="https://doi.org/10.1016/j.jmaa.2021.125407">No dimension reduction for doubling subsets of $\ell_q$ when q > 2 revisited</a>, Journal of Mathematical Analysis and Applications, vol. 504, no. 2. Elsevier BV, p. 125407, Dec. 2021.
%H A365273 S. J. Dilworth, D. Kutzarova, and M. I. Ostrovskii, <a href="https://doi.org/10.4153/S0008414X19000087">Lipschitz-free Spaces on Finite Metric Spaces</a>, Canadian Journal of Mathematics, vol. 72, no. 3. Canadian Mathematical Society, pp. 774-804, Feb. 13, 2019.
%H A365273 Stephen J. Dilworth, Denka Kutzarova, and Mikhail I. Ostrovskii, <a href="https://arxiv.org/abs/2007.07949">Analysis on Laakso graphs with application to the structure of transportation cost spaces</a>, arXiv:2007.07949 [math.FA], 2020-2021. See drawing of L_1 on page 4.
%H A365273 S. J. Dilworth, D. Kutzarova, and M. I. Ostrovskii, <a href="https://doi.org/10.1007/s11117-021 00821-w">Analysis on Laakso graphs with application to the structure of transportation cost spaces</a>, Positivity 25, 1403-1435 (2021).
%H A365273 S. J. Dilworth, D. Kutzarova, and S. Stankov, <a href="https://doi.org/10.1007/s43037-022-00212-7">Metric embeddings of Laakso graphs into Banach spaces.</a> Banach J. Math. Anal. 16, 60 (2022).
%H A365273 T. Laakso, <a href="https://doi.org/10.1007/s000390050003">Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality</a>, GAFA, Geom. funct. anal. 10, 111-123 (2000).
%H A365273 U. Lang and C. Plaut, <a href="https://doi.org/10.1023/A:1012093209450">Bilipschitz Embeddings of Metric Spaces into Space Forms</a>, Geometriae Dedicata 87, 285-307 (2001).
%H A365273 A. Margaris and J. C. Robinson, <a href="https://doi.org/10.5802/crmath.70">Some comments on Laakso graphs and sets of differences</a>, Comptes Rendus. Mathématique, vol. 358, no. 4. Cellule MathDoc/CEDRAM, pp. 515-521, Jul. 28, 2020.
%H A365273 MathOverflow, <a href="https://stackoverflow.com/questions/76511046/implementation-of-the-laakso-graph-in-python">Implementation of the "Laakso Graph" in Python</a>, 2023.
%H A365273 O. Neiman, <a href="https://doi.org/10.1007/s00224-014-9567-3">Low Dimensional Embeddings of Doubling Metrics</a>, Theory Comput Syst 58, 133-152 (2016).
%H A365273 Th. Schlumprecht and G. Tresch, <a href="https://arxiv.org/abs/2306.06222">“Stochastic Embeddings of Graphs into Trees.”</a> arXiv preprint arXiv:2306.06222 [math.CO], 2023.
%H A365273 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-6).
%F A365273 a(n) = a(n-1) + 4*6^(n-1).
%F A365273 a(n) = (2/5) * (2*6^n+3). - _Christian Krause_, Sep 30 2023
%e A365273 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.
%t A365273 LinearRecurrence[{7,-6},{6,30},30] (* _Paolo Xausa_, Oct 16 2023 *)
%Y A365273 Equals twice A152596.
%K A365273 nonn,easy
%O A365273 1,1
%A A365273 _Ken McCabe_, Aug 30 2023