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%I #81 Mar 17 2024 17:42:55
%S 6,24,96,384,1536,6144,24576,98304,393216,1572864,6291456,25165824,
%T 100663296,402653184,1610612736,6442450944,25769803776,103079215104,
%U 412316860416,1649267441664,6597069766656,26388279066624,105553116266496,422212465065984
%N a(n) = 6*4^n.
%C From _Peter M. Chema_, Mar 02 2017: (Start)
%C Number of rods (line segments) required to make a Sierpinski tetrahedron of side length 2^n.
%C Also equals the number of balls (vertices) in a Sierpinski tetrahedron of side length 2^n+1 minus the number of balls in a Sierpinski tetrahedron of side length 2^n (the first difference in the tetrix numbers). See formula. (End)
%C Equivalently, the number of edges in the (n+1)-Sierpinski tetrahedron graph. - _Eric W. Weisstein_, Aug 17 2017
%C These numbers a(n) together with the 13 numbers from A337217 give the positive integers m represented uniquely by the ternary form x^2 + y^2 + 2*z^2, with integers 0 <= x <= y and 0 <= z. This is theorem 2.1 of Kaplansky, p. 87 with proof on p. 90. - _Wolfdieter Lang_, Aug 20 2020
%C a(n) is also the domination number of the (n+3)-Sierpinski tetrahedron graph. - _Eric W. Weisstein_, Sep 13 2021
%D Irving Kaplansky, Integers Uniquely Represented by Certain Ternary Forms, in "The Mathematics of Paul Erdős I", Ronald. L. Graham and Jaroslav Nešetřil (Eds.), Springer, 1997, pp. 86 - 94.
%H Vincenzo Librandi, <a href="/A002023/b002023.txt">Table of n, a(n) for n = 0..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiTetrahedronGraph.html">Sierpinski Tetrahedron Graph</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F From _Philippe Deléham_, Nov 23 2008: (Start)
%F a(n) = 4*a(n-1) for n > 0, a(0)=6.
%F G.f.: 6/(1-4*x). (End)
%F a(n) = 3*A004171(n). - _R. J. Mathar_, Mar 08 2011
%F From _Peter M. Chema_, Mar 03 2017: (Start)
%F a(n) = A283070(n+1) - A283070(n).
%F a(n) = A004171(n+1) - A004171(n). (End)
%F E.g.f.: 6*exp(4*x). - _G. C. Greubel_, Aug 17 2017
%t 6*4^Range[0, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jun 09 2011 *)
%t Table[6 4^n, {n, 0, 20}] (* _Eric W. Weisstein_, Aug 17 2017 *)
%t LinearRecurrence[{4}, {6}, 20] (* _Eric W. Weisstein_, Aug 17 2017 *)
%t CoefficientList[Series[6/(1 - 4 x), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 17 2017 *)
%t NestList[4#&,6,30] (* _Harvey P. Dale_, Mar 17 2024 *)
%o (Magma) [6*4^n: n in [0..30]]; // _Vincenzo Librandi_, May 16 2011
%o (PARI) a(n)=6<<(2*n) \\ _Charles R Greathouse IV_, Apr 17 2012
%Y Cf. A283070 (vertex count).
%Y Cf. A004171.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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