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

Author

Peter M. Chema, Feb 28 2017

Keywords

• nonn
• easy

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).

Crossrefs

Subsequence of A016957.

Cf. A052539, A279511, A279512.

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

Cf. A002023 (edge count).

Programs

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 *)

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

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

Extensions

Entry revised by Editors of OEIS, Mar 01 2017