A061777 Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives total population of triangles at n-th generation.

1, 4, 10, 22, 40, 70, 112, 178, 268, 406, 592, 874, 1252, 1822, 2584, 3730, 5260, 7558, 10624, 15226, 21364, 30574, 42856, 61282, 85852, 122710, 171856, 245578, 343876, 491326, 687928, 982834, 1376044, 1965862, 2752288, 3931930, 5504788

Offset: 0

Author

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

Keywords

• nonn
• nice
• easy

Comments

From the definition, assign label value "1" to an origin triangle; at n-th generation add a triangle at each vertex. Each non-overlapping triangle will have the same label value as that of the predecessor triangle to which it is connected; for the overlapping ones, the label value will be the sum of the label values of predecessors. a(n) is the sum of all label values at the n-th generation. The triangle count is A005448. See illustration. For n >= 1, (a(n) - a(n-1))/3 is A027383. - Kival Ngaokrajang, Sep 05 2014

References

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

Links

• Kival Ngaokrajang, Illustration of initial terms
• 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 (2,1,-4,2).

Crossrefs

Partial sums of A061776.

Cf. A005448, A027383.

Programs

Maple

seq(`if`(n::even, 21*2^(n/2) - 6*n-20, 30*2^((n-1)/2)-6*n-20),n=0..100); # Robert Israel, Sep 14 2014

Mathematica

Table[If[EvenQ[n],21 2^(n/2)-6n-20,30 2^((n-1)/2)-6(n-1)-26],{n,0,40}] (* Harvey P. Dale, Nov 06 2011 *)

PARI

a(n)=if(n%2, 30, 21)<<(n\2) - 6*n - 20 \\ Charles R Greathouse IV, Sep 19 2014

Formula

From Colin Barker, May 08 2012: (Start)

a(n) = 21*2^(n/2) - 6*n - 20 if n is even.

a(n) = 30*2^((n-1)/2) - 6*(n - 1) - 26 if n is odd.

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + 2*a(n-4).

G.f.: (1 + 2*x)*(1 + x^2)/((1 - x)^2*(1 - 2*x^2)). (End)

From Robert Israel, Sep 14 2014: (Start)

a(n) = -20 - 6*n + (21 + 15*sqrt(2))*sqrt(2)^(n-2) + (21 - 15*sqrt(2))*(-sqrt(2))^(n-2).

a(n) = 2*a(n-2) + ((3*n-2)/(3*n-5))*(a(n-1)-2*a(n-3)). (End)

E.g.f.: 21*cosh(sqrt(2)*x) + 15*sqrt(2)*sinh(sqrt(2)*x) - 2*exp(x)*(10 + 3*x). - Stefano Spezia, Aug 13 2022

Extensions

Corrected by T. D. Noe, Nov 08 2006