A002023 a(n) = 6*4^n.

6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A283070 (vertex count).

Cf. A004171.

Programs

Magma

[6*4^n: n in [0..30]]; // Vincenzo Librandi, May 16 2011

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 *)

PARI

a(n)=6<<(2*n) \\ Charles R Greathouse IV, Apr 17 2012

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)

a(n) = A283070(n+1) - A283070(n).

a(n) = A004171(n+1) - A004171(n). (End)

E.g.f.: 6*exp(4*x). - G. C. Greubel, Aug 17 2017