A097108 If a geodesic dome is made by dividing each triangle of an icosahedron into n^2 identical equilateral triangles and the vertices of those newly created triangles are pushed out from the center to lie on the surface of the sphere in which the icosahedron is inscribed, then this sequence gives the number of different strut lengths that are required to build the dome.

1, 2, 3, 6, 9, 9, 16, 20, 18, 30, 36, 30, 49, 56, 45, 72, 81, 63, 100, 110, 84, 132, 144, 108, 169, 182, 135, 210, 225, 165, 256, 272, 198, 306, 324, 234, 361, 380, 273, 420, 441, 315, 484, 506, 360, 552, 576, 408, 625, 650, 459, 702, 729, 513, 784, 812, 570, 870

Offset

1,2

Comments

From Gonzalo Rodríguez Whipple, May 30 2010: (Start)

The sequence consists of two series, one of which consists of two subseries.

1. If n is a multiple of 3: n^2/6+n/2.

2.1 If n=1 or n+1 is a multiple of 6 or n-1 is a multiple of 6: n^2/4+n/2+1/4.

2.2 Otherwise: n^2/4+n/2.

This means that frequencies (n) that are multiples of 3 assure a higher symmetry and need a smaller number of different strut lengths. (End)

Links

Table of n, a(n) for n=1..58.

Tara Landry, Desert Domes.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,0,0,0,-2,0,0,1).

Formula

Satisfies a linear recurrence with characteristic polynomial (1+x^3)(1-x^3)^3.

From Gonzalo Rodríguez Whipple, May 30 2010: (Start)

a(n) = if(n mod 3, (n^2)/6+n/2, 0)

+ if(((n+1) mod 6) or ((n-1) mod 6), (n^2)/4+n/2+1/4), 0)

+ if(((n+2) mod 6) or ((n-2) mod 6), (n^2)/4+n/2), 0). (End)

G.f.: -x*(2*x^7+4*x^6+3*x^5+5*x^4+4*x^3+3*x^2+2*x+1)/((x-1)^3*(x+1)*(x^2-x+1)*(x^2+x+1)^3). [Colin Barker, Oct 21 2012]

a(n) = ((2*n^2+6*n)*(1-(n^2 mod 3))+3*(n+1)^2*(n^2 mod (5-(-1)^n)/2)+(3*n^2 +6*n)*((n+3)^2 mod (5+(-1)^n)/2))/12. - Wesley Ivan Hurt, Mar 11 2015

A045943(n) = a(3*(n-1)), n<>1. - Gonzalo Rodríguez Whipple, May 30 2010

Example

a(4) = 6 since we can build a "4V" dome of radius 1 using 30 struts of length 0.25318, 30 struts of length 0.29453, 70 of length 0.31287, 30 of length 0.32492 and 30 of length 0.29859. The number 6 indicates the number of different jig settings we'd have to use to manufacture all the struts for this dome.
a(299)=22500, a(300)=15150, a(301)=22801. [Gonzalo Rodríguez Whipple, May 30 2010]

Mathematica

Table[If[Divisible[n, 3], (n^2)/6 + n/2, 0] + If[Divisible[n + 1, 6] || Divisible[n - 1, 6], (n^2)/4 + n/2 + 1/4, 0] + If[Divisible[n + 2, 6] || Divisible[n - 2, 6], (n^2)/4 + n/2, 0], {n, 100}] (* Gonzalo Rodríguez Whipple, May 30 2010 *)

Crossrefs

Cf. A045943.

Sequence in context: A309008 A249182 A189968 * A140783 A094351 A061910

Adjacent sequences: A097105 A097106 A097107 * A097109 A097110 A097111

Keyword

nonn,easy

Author

Tom Davis (tomrdavis(AT)earthlink.net), Sep 15 2004

Status

approved