A279512 - OEIS

%I #34 Jun 19 2021 10:57:38

%S 6,19,85,457,2641,15649,93505,560257,3360001,20156929,120935425,

%T 725600257,4353576961,26121412609,156728377345,940370067457,

%U 5642220011521,33853319282689,203119914123265,1218719481593857,7312316883271681,43873901287047169,263243407697117185

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

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

%H Colin Barker, <a href="/A279512/b279512.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/N-flake#Octahedron_flake">Octahedron flake</a> and <a href="https://en.wikipedia.org/wiki/Sierpinski_triangle#Analogues_in_higher_dimensions">Sierpinski tetrahedron</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-20,12).

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

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

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

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

%F From _Colin Barker_, Dec 15 2016: (Start)

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

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

%F (End)

%t 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 *)

%o (PARI) Vec((6 - 35*x + 34*x^2) / ((1 - x)*(1 - 2*x)*(1 - 6*x)) + O(x^30)) \\ _Colin Barker_, Dec 15 2016

%o (Python)

%o def a(n): return 2*6**n + 3*2**n + 1

%o print([a(n) for n in range(23)]) # _Michael S. Branicky_, Jun 19 2021

%Y Cf. A005900, A047999, A279511.

%K nonn,easy

%O 0,1

%A _Steven Beard_, Dec 14 2016

%E Incorrect terms corrected by _Colin Barker_, Dec 15 2016