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%I #47 Aug 29 2021 22:17:00
%S 4,10,34,130,514,2050,8194,32770,131074,524290,2097154,8388610,
%T 33554434,134217730,536870914,2147483650,8589934594,34359738370,
%U 137438953474,549755813890,2199023255554,8796093022210,35184372088834,140737488355330,562949953421314
%N Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.
%C Number of vertices required to make a Sierpinski tetrahedron or tetrix of side length 2^n. The sum of the vertices (balls) plus line segments (rods) of one tetrix equals the vertices of its larger, adjacent iteration. See formula.
%C Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - _Eric W. Weisstein_, Aug 17 2017
%C Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - _Eric W. Weisstein_, Aug 29 2021
%C Final digit alternates 4 and 0.
%H Colin Barker, <a href="/A283070/b283070.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiTetrahedronGraph.html">Sierpinski Tetrahedron Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tetrix.html">Tetrix</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCount.html">Vertex Count</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).
%F a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
%F a(n+1) = a(n) + A002023(n).
%F a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).
%F E.g.f.: 2*(exp(4*x) + exp(x)). - _G. C. Greubel_, Aug 17 2017
%t Table[2 4^n + 2, {n, 0, 30}] (* _Bruno Berselli_, Feb 28 2017 *)
%t 2 (4^Range[0, 20] + 1) (* _Eric W. Weisstein_, Aug 17 2017 *)
%t LinearRecurrence[{5, -4}, {4, 10}, 20] (* _Eric W. Weisstein_, Aug 17 2017 *)
%t CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 17 2017 *)
%o (PARI) a(n)=2*4^n+2 \\ _Charles R Greathouse IV_, Feb 28 2017
%o (PARI) Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ _Colin Barker_, Mar 02 2017
%o (Python)
%o def a(n): return 2*4**n + 2
%o print([a(n) for n in range(25)]) # _Michael S. Branicky_, Aug 29 2021
%Y Subsequence of A016957.
%Y Cf. A052539, A279511, A279512.
%Y First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
%Y Cf. A002023 (edge count).
%K nonn,easy
%O 0,1
%A _Peter M. Chema_, Feb 28 2017
%E Entry revised by Editors of OEIS, Mar 01 2017
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