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

0,1

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

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)

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

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

Cf. A283070 (vertex count).

Cf. A004171.

Sequence in context: A344039 A253101 A169759 * A164908 A290911 A037505

Adjacent sequences: A002020 A002021 A002022 * A002024 A002025 A002026

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved