A152596 -id:A152596 - cp's OEIS frontend

A152599 a(n) = 10a(n-1) - 12a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

1, 4, 28, 232, 1984, 17056, 146752, 1262848, 10867456, 93520384, 804794368, 6925699072, 59599458304, 512886194176, 4413668442112, 37982050091008, 326856479604736, 2812780194955264, 24205524194295808, 208301879603494912, 1792552505703399424, 15425902501792055296

Offset: 0

Philippe Deléham, Dec 09 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-12).

Cf. A052961, A006012, A147703, A152596.

LinearRecurrence[{10, -12}, {1, 4}, 25] (* Paolo Xausa, Jan 19 2024 *)

G.f.: (1-6*x)/(1-10*x+12*x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*3^(n-k).
a(n) = 2^n*A052961(n). - R. J. Mathar, Jun 14 2016

A365273 Number of vertices in the Laakso graph of order n.

Original entry on oeis.org

6, 30, 174, 1038, 6222, 37326, 223950, 1343694, 8062158, 48372942, 290237646, 1741425870, 10448555214, 62691331278, 376147987662, 2256887925966, 13541327555790, 81247965334734, 487487792008398, 2924926752050382

Offset: 1

Ken McCabe, Aug 30 2023

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.

			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.

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).

Equals twice A152596.

LinearRecurrence[{7,-6},{6,30},30] (* Paolo Xausa, Oct 16 2023 *)

a(n) = a(n-1) + 4*6^(n-1).

a(n) = (2/5) * (2*6^n+3). - Christian Krause, Sep 30 2023

A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 7, 20, 27, 13, 1, 9, 35, 73, 80, 34, 1, 11, 54, 151, 252, 234, 89, 1, 13, 77, 269, 597, 837, 677, 233, 1, 15, 104, 435, 1199, 2225, 2702, 1941, 610, 1, 17, 135, 657, 2158, 4956, 7943, 8533, 5523, 1597

Offset: 0

Philippe Deléham, Nov 06 2011

Mirror image of triangle in A147703.

			Triangle begins:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  9,  5;
  1,  7, 20, 27, 13;
  1,  9, 35, 73, 80, 34;

Cf. A001519, A059502, A000012, A005408, A014107.

Sum_{k=0..n} T(n,k)*x^k = A152620(n), A152594(n), A000007(n), A000012(n), A006012(n), A152596(n), A152599(n) for x=-3,-2,-1,0,1,2,3 respectively.
T(n,n) = A001519(n).
G.f.: (1-2y*x)/(1-(1+3y)*x+y*(1+y)*x^2).

cp's OEIS Frontend

A152599 a(n) = 10a(n-1) - 12a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

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A365273 Number of vertices in the Laakso graph of order n.

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A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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Formula

cp's OEIS Frontend

A152599 a(n) = 10*a(n-1) - 12*a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A365273 Number of vertices in the Laakso graph of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A152599 a(n) = 10a(n-1) - 12a(n-2) for n > 1; a(0) = 1, a(1) = 4 .