A061776 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.

1, 3, 6, 12, 18, 30, 42, 66, 90, 138, 186, 282, 378, 570, 762, 1146, 1530, 2298, 3066, 4602, 6138, 9210, 12282, 18426, 24570, 36858, 49146, 73722, 98298, 147450, 196602, 294906, 393210, 589818, 786426, 1179642, 1572858, 2359290, 3145722, 4718586, 6291450

Offset

0,2

Comments

Number of 3-colorings of the (n,2)-Turán graph. - Alois P. Heinz, Jun 07 2024

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. Reed, The Lemming Simulation Problem, 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]

Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Formula

Explicit formula given in Maple line.

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

a(n) = 3*A027383(n-1) for n>0, a(0)=1. - Bruno Berselli, May 08 2012

Maple

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

Mathematica

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 *)
CoefficientList[Series[(1 + 2 x) (1 + x^2) / ((1 - x) (1 - 2 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
LinearRecurrence[{1, 2, -2}, {1, 3, 6, 12}, 40] (* Harvey P. Dale, Mar 27 2019 *)

Prog

(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

Crossrefs

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

Cf. A266972.

Sequence in context: A242477 A006156 A171370 * A356020 A341316 A298029

Adjacent sequences: A061773 A061774 A061775 * A061777 A061778 A061779

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, R. K. Guy, Jun 23 2001

Status

approved