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A160249
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Triangle of "less regular" face-centered-cubic sphere pack cuboctahedron numbers read by rows.
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1
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1, 4, 13, 10, 28, 55, 20, 50, 92, 147, 35, 80, 140, 216, 309, 56, 119, 200, 300, 420, 561, 84, 168, 273, 400, 550, 724, 923, 120, 228, 360, 517, 700, 910, 1148, 1415, 165, 300, 462, 652, 871, 1120, 1400, 1712, 2057
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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These solids each have 6 same rectangles, if one includes the first column (tetrahedra), where the rectangle is 1*(row number). They each have 2 tetrahedral quartets of same equilateral triangles, if one includes again the first column, for whom one or both quartets may be a single unit.
The first column is the series of tetrahedral numbers, (essentially A000292) and represents increasing edge length of one quartet of equilateral triangles. The column number represents the other quartet. The 2 quartet edges are interchangeable, because this triangle array is one half of a square table reflectable about the diagonal.
So the final column of each row represents a point where the 2 quartets' edge lengths are equal and the result is a regular cuboctahedron (A005902).
It seems remarkable that 24 of these shapes have even-hundred quantities of 10,000 or less, including 10,000 itself! This is far more than in any other figurate number series I've encountered. Additionally, 300 occurs twice, at 2,9 and 4,6 as does 9100, at 7,24 and 13,16. It suggests to me that this class of shapes may be useful in illustrating large numbers attractively.
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REFERENCES
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Main Title: Polyhedra primer / Peter Pearce and Susan Pearce. Published/Created: New York : Van Nostrand Reinhold, c1978. Description: viii, 134 p. : ill. ; 24 cm. ISBN: 0442264968
Main Title: The book of numbers / John H. Conway, Richard K. Guy. Published/Created: New York, NY : Copernicus c1996. Description: ix, 310 p. : ill. (some col.) ; 24 cm. ISBN: 038797993X
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LINKS
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FORMULA
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v=(y^3+x^3+9*y^2*x+9*y*x^2-6*y^2-6*x^2-18*y*x+11*y+11*x-6)/6
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PROG
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(Excel) Paste the following formula into cell C3, and fill down and right to desired table size. All volumes 10, 000 and under are covered by column Q and row 38. The cells above the diagonal can be cleared since they are duplicates of content below the diagonal.
=((ROW()-2)^3+(COLUMN()-2)^3+9*(ROW()-2)^2*(COLUMN()-2)+9*(ROW()-2)*(COLUMN()-2)^2-6*(ROW()-2)^2-6*(COLUMN()-2)^2-18*(ROW()-2)*(COLUMN()-2)+11*(ROW()-2)+11*(COLUMN()-2)-6)/6
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CROSSREFS
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KEYWORD
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AUTHOR
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Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 05 2009, May 11 2009
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EXTENSIONS
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Improvement of the definition's precision by Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 19 2009
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STATUS
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approved
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