OFFSET
0,1
COMMENTS
From Peter M. Chema, Mar 02 2017: (Start)
Number of rods (line segments) required to make a Sierpinski tetrahedron of side length 2^n.
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)
Equivalently, the number of edges in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017
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
a(n) is also the domination number of the (n+3)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 13 2021
REFERENCES
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.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 17, 26.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Domination Number
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Index entries for linear recurrences with constant coefficients, signature (4).
FORMULA
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 4*a(n-1) for n > 0, a(0)=6.
G.f.: 6/(1-4*x). (End)
a(n) = 3*A004171(n). - R. J. Mathar, Mar 08 2011
From Peter M. Chema, Mar 03 2017: (Start)
E.g.f.: 6*exp(4*x). - G. C. Greubel, Aug 17 2017
MATHEMATICA
6*4^Range[0, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[6 4^n, {n, 0, 20}] (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{4}, {6}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[6/(1 - 4 x), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)
NestList[4#&, 6, 30] (* Harvey P. Dale, Mar 17 2024 *)
PROG
(Magma) [6*4^n: n in [0..30]]; // Vincenzo Librandi, May 16 2011
(PARI) a(n)=6<<(2*n) \\ Charles R Greathouse IV, Apr 17 2012
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved