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Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives number of triangles in n-th generation.
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%I #40 Jun 07 2024 11:23:05

%S 1,3,6,12,18,30,42,66,90,138,186,282,378,570,762,1146,1530,2298,3066,

%T 4602,6138,9210,12282,18426,24570,36858,49146,73722,98298,147450,

%U 196602,294906,393210,589818,786426,1179642,1572858,2359290,3145722,4718586,6291450

%N Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives number of triangles in n-th generation.

%C Number of 3-colorings of the (n,2)-Turán graph. - _Alois P. Heinz_, Jun 07 2024

%D R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.

%H Vincenzo Librandi, <a href="/A061776/b061776.txt">Table of n, a(n) for n = 0..1000</a>

%H R. Reed, <a href="/A005448/a005448_1.pdf">The Lemming Simulation Problem</a>, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).

%F Explicit formula given in Maple line.

%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3) for n>3. G.f.: (1+2*x)*(1+x^2)/((1-x)*(1-2*x^2)). - _Colin Barker_, May 08 2012

%F a(n) = 3*A027383(n-1) for n>0, a(0)=1. - _Bruno Berselli_, May 08 2012

%p A061776 := proc(n) if n mod 2 = 0 then 6*(2^(n/2)-1); else 3*(2^((n-1)/2)-1)+3*(2^((n+1)/2)-1); fi; end; # for n >= 1

%t a[0]=1; a[n_/;EvenQ[n]]:=6*(2^(n/2)-1); a[n_/;OddQ[n]] := 3*(2^((n-1)/2)-1) + 3*(2^((n+1)/2)-1); a /@ Range[0, 37] (* _Jean-François Alcover_, Apr 22 2011, after Maple program *)

%t CoefficientList[Series[(1 + 2 x) (1 + x^2) / ((1 - x) (1 - 2 x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)

%t LinearRecurrence[{1,2,-2},{1,3,6,12},40] (* _Harvey P. Dale_, Mar 27 2019 *)

%o (PARI) a(n)=([0,1,0; 0,0,1; -2,2,1]^n*[1;3;6])[1,1] \\ _Charles R Greathouse IV_, Feb 19 2017

%Y A061777 gives total population of triangles at n-th generation.

%Y Cf. A266972.

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_, _R. K. Guy_, Jun 23 2001