A279512 Sierpinski octahedron numbers a(n) = 2*6^n + 3*2^n + 1.

6, 19, 85, 457, 2641, 15649, 93505, 560257, 3360001, 20156929, 120935425, 725600257, 4353576961, 26121412609, 156728377345, 940370067457, 5642220011521, 33853319282689, 203119914123265, 1218719481593857, 7312316883271681, 43873901287047169, 263243407697117185

Offset: 0

Author

Steven Beard, Dec 14 2016

Keywords

• nonn
• easy

Comments

Sierpinski recursion applied to octahedron. Cf. A279511 for square pyramids.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Wikipedia, Octahedron flake and Sierpinski tetrahedron.
• Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Crossrefs

Cf. A005900, A047999, A279511.

Programs

Mathematica

LinearRecurrence[{9, -20, 12}, {6, 19, 85}, 50] (* or *) Table[2*6^n + 3*2^n + 1, {n,0,50}] (* G. C. Greubel, Dec 22 2016 *)

PARI

Vec((6 - 35*x + 34*x^2) / ((1 - x)*(1 - 2*x)*(1 - 6*x)) + O(x^30)) \\ Colin Barker, Dec 15 2016

Python

def a(n): return 2*6**n + 3*2**n + 1
print([a(n) for n in range(23)]) # Michael S. Branicky, Jun 19 2021

Formula

a(n) = 3*2^n + 2^(n+1)*3^n + 1.

a(n) = 6a(n-1) - 6(2^n+1) + 1.

a(n) = 6a(n-1) - (3*2^(n+1) + 5).

a(n) = 2*6^n + 3*2^n + 1.

From Colin Barker, Dec 15 2016: (Start)

a(n) = 9*a(n-1) - 20*a(n-2) + 12*a(n-3) for n>2.

G.f.: (6 - 35*x + 34*x^2) / ((1 - x)*(1 - 2*x)*(1 - 6*x)).

(End)

Extensions

Incorrect terms corrected by Colin Barker, Dec 15 2016