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 A100774 #84 Feb 16 2025 08:32:55
%S A100774 0,4,16,52,160,484,1456,4372,13120,39364,118096,354292,1062880,
%T A100774 3188644,9565936,28697812,86093440,258280324,774840976,2324522932,
%U A100774 6973568800,20920706404,62762119216,188286357652,564859072960,1694577218884
%N A100774 a(n) = 2*(3^n - 1).
%C A100774 a(n) is the number of steps which are made when generating all n-step nonreversing random walks that begin in a fixed point P on a two-dimensional square lattice. To make one step means to move along one edge on the lattice.
%C A100774 These are also the first local maxima reached in the Collatz trajectories of 2^n - 1. - _David Rabahy_, Oct 30 2017
%C A100774 Also the graph diameter of the n-Sierpinski carpet graph. - _Eric W. Weisstein_, Mar 13 2018
%C A100774 a(n) is the number of edge covers of F_{n,2}, which has adjacent vertices u and w, and n vertices each adjacent to both u and w. An edge cover is a subset of the edges where each vertex is adjacent to at least one vertex. To cover each of the n vertices v_i, we need to have at least the edge uv_i or wv_i or both, giving us three choices for each. We can then add the edge uw or not, which is 2*3^n choices. But we need to remove the case where all uv_i's were chosen and uw not chosen, and all ww_i's were chosen and uw not chosen. - _Feryal Alayont_, Jun 17 2024
%H A100774 Vincenzo Librandi, <a href="/A100774/b100774.txt">Table of n, a(n) for n = 0..196</a>
%H A100774 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDiameter.html">Graph Diameter</a>
%H A100774 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiCarpetGraph.html">Sierpinski Carpet Graph</a>
%H A100774 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A100774 a(n) = 2*(3^n - 1).
%F A100774 a(n) = 4*Sum_{i=0..n-1} 3^i.
%F A100774 a(n) = 4*A003462(n).
%F A100774 a(n) = A048473(n) - 1. - _Paul Curtz_, Jan 19 2009
%F A100774 G.f.: 4*x/((1-x)*(1-3*x)). - _Eric W. Weisstein_, Mar 13 2018
%F A100774 a(n) = 4*a(n-1) - 3*a(n-2). - _Eric W. Weisstein_, Mar 13 2018
%F A100774 From _Elmo R. Oliveira_, Dec 06 2023: (Start)
%F A100774 a(n) = 2*A024023(n).
%F A100774 a(n) = 3*a(n-1) + 4 for n>0.
%F A100774 E.g.f.: 2*(exp(3*x) - exp(x)). (End)
%t A100774 Table[2 (3^n - 1), {n, 0, 24}] (* _Alonso del Arte_, Nov 08 2012 *)
%t A100774 2 (3^Range[0, 20] - 1) (* _Eric W. Weisstein_, Mar 13 2018 *)
%t A100774 LinearRecurrence[{4, -3}, {4, 16}, {0, 20}] (* _Eric W. Weisstein_, Mar 13 2018 *)
%t A100774 CoefficientList[Series[4 x/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 13 2018 *)
%o A100774 (Magma) [2*(3^n - 1): n in [0..25] ]; // _Vincenzo Librandi_, Apr 30 2011
%o A100774 (Maxima) A100774(n):=2*(3^n - 1)$
%o A100774 makelist(A100774(n),n,0,30); /* _Martin Ettl_, Nov 09 2012 */
%o A100774 (PARI) a(n)=2*3^n-2 \\ _Charles R Greathouse IV_, Nov 09 2012
%Y A100774 Cf. A003462, A007051, A024023, A029858, A034472, A048473, A058481, A067771, A079004, A115099, A134931.
%K A100774 easy,nonn
%O A100774 0,2
%A A100774 Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005