A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154, 8388610, 33554434, 134217730, 536870914, 2147483650, 8589934594, 34359738370, 137438953474, 549755813890, 2199023255554, 8796093022210, 35184372088834, 140737488355330, 562949953421314

Offset

0,1

Comments

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.

Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017

Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 29 2021

Final digit alternates 4 and 0.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Independence Number

Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph

Eric Weisstein's World of Mathematics, Tetrix

Eric Weisstein's World of Mathematics, Vertex Count

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Formula

G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).

a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.

a(n+1) = a(n) + A002023(n).

a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).

E.g.f.: 2*(exp(4*x) + exp(x)). - G. C. Greubel, Aug 17 2017

Mathematica

Table[2 4^n + 2, {n, 0, 30}] (* Bruno Berselli, Feb 28 2017 *)
2 (4^Range[0, 20] + 1) (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{5, -4}, {4, 10}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)

Prog

(PARI) a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017
(PARI) Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017
(Python)
def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021

Crossrefs

Subsequence of A016957.

Cf. A052539, A279511, A279512.

First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.

Cf. A002023 (edge count).

Sequence in context: A006343 A149173 A149174 * A222631 A030003 A339845

Adjacent sequences: A283067 A283068 A283069 * A283071 A283072 A283073

Keyword

nonn,easy

Author

Peter M. Chema, Feb 28 2017

Extensions

Entry revised by Editors of OEIS, Mar 01 2017

Status

approved