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A097108
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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.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Contribution from Gonzalo Rodriguez W (gonzalorodriguezarquitecto(AT)gmail.com), May 30 2010: (Start)
The sequence consists of two series one of which consists of two subseries.
1. If n is multiple of 3: n^2/6+n/2
2.1 If n=1 or n+1 is multiple of 6 or n-1 is 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 multiple of 3 assure a higher symmetry
and need a smaller number of different strut lengths. (End)
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LINKS
| Tara Landry, Desert Domes.
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FORMULA
| Satisfies a linear recurrence with characteristic polynomial (1+x^3)(1-x^3)^3.
Contribution from Gonzalo Rodriguez W (gonzalorodriguezarquitecto(AT)gmail.com), 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)
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EXAMPLE
| a(4) = 6 since we can build a "4V" dome of radius 1 using 30 struts of length .25318, 30 struts of length .29453, 70 of length .31287, 30 of length .32492 and 30 of length .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 [From Gonzalo Rodriguez W (gonzalorodriguezarquitecto(AT)gmail.com), May 30 2010]
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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}] [From Gonzalo Rodriguez W (gonzalorodriguezarquitecto(AT)gmail.com), May 30 2010]
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CROSSREFS
| A045943(n)=A097108(3*(n-1)), n<>1 [From Gonzalo Rodriguez W (gonzalorodriguezarquitecto(AT)gmail.com), May 30 2010]
Sequence in context: A087494 A021426 A189968 * A140783 A094351 A061910
Adjacent sequences: A097105 A097106 A097107 * A097109 A097110 A097111
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KEYWORD
| nonn
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AUTHOR
| Tom Davis (tomrdavis(AT)earthlink.net), Sep 15 2004
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