cp's OEIS Frontend

A061709 Consider a (hollow) triangle with n cells on each edge, for a total of 3(n-1) cells if n>1, or 1 cell if n=1; a(n) is number of ways of labeling cells with 0's and 1's; triangle may be rotated and turned over.

%I A061709 #15 Aug 18 2024 02:01:24
%S A061709 1,4,20,104,752,5600,44224,350592,2800384,22377984,178990080,
%T A061709 1431721984,11453509632,91626496000,733009854464,5864066220032,
%U A061709 46912512917504,375300002545664,3002399885885440,24019198281252864,192153585175232512,1537228674957312000
%N A061709 Consider a (hollow) triangle with n cells on each edge, for a total of 3(n-1) cells if n>1, or 1 cell if n=1; a(n) is number of ways of labeling cells with 0's and 1's; triangle may be rotated and turned over.
%H A061709 Colin Barker, <a href="/A061709/b061709.txt">Table of n, a(n) for n = 1..1000</a>
%H A061709 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-8,-80,128).
%F A061709 a(n) = (1/6)*(2^(3*(n-1))+2^n+3*2^(floor((3*n-1)/2))) for n>1.
%F A061709 a(2)=4, a(3)=20, a(4)=104, a(5)=752, a(n)=10*a(n-1)-8*a(n-2)- 80*a(n-3)+ 128*a(n-4). - _Harvey P. Dale_, Apr 22 2013
%F A061709 G.f.: -x*(64*x^4+16*x^3-12*x^2-6*x+1) / ((2*x-1)*(8*x-1)*(8*x^2-1)). - _Colin Barker_, Mar 17 2015
%e A061709 a(2) = 4, the labelings being {000}, {001}, {011}, {111}. Some of the 20 solutions for n=3 are as follows:
%e A061709 ..0......1.......0......1.......1.......1.......0
%e A061709 .0.0....0.0.....1.0....1.0.....0.0.....0.0.....1.1
%e A061709 0.0.0..0.0.0...0.0.0..0.0.0...1.0.0...0.1.0...0.0.0
%e A061709 The first solution for n = 4 is
%e A061709 ...0
%e A061709 ..0.0
%e A061709 .0...0
%e A061709 0.0.0.0
%t A061709 Join[{1},Table[((2^(3(n-1)))+2^n+3*2^Floor[(3n-1)/2])/6,{n,2,30}]] (* or *) Join[{1},LinearRecurrence[{10,-8,-80,128},{4,20,104,752},30]] (* _Harvey P. Dale_, Apr 22 2013 *)
%o A061709 (PARI) Vec(-x*(64*x^4+16*x^3-12*x^2-6*x+1)/((2*x-1)*(8*x-1)*(8*x^2-1)) + O(x^100)) \\ _Colin Barker_, Mar 17 2015
%Y A061709 Cf. A061348.
%K A061709 nonn,easy,nice
%O A061709 1,2
%A A061709 _N. J. A. Sloane_, Jun 20 2001