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A100012
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Iterated hypericosahedron numbers, starting with hypericosahedron(2) = 120. The hypericosahedron numbers are 4-dimensional figurate numbers based on the 600-cell regular convex polytope, also known as the hexacosichoron or hypericosahedron. This sequence is hypericosahedron(2), hypericosahedron(hypericosahedron(2)) and so on recursively.
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0
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2, 120, 4930988840, 14287387711051307292599794275187472361080, 1006996758347337753704336981217909980511370171347746397003380972662176429495263574435591467113712332335247912386392833953584459582398757124798954874818723391550760
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| This need not start at hypericosahedron(2) = 120. For example, starting at a(0) = 7, which is not a hypericosahedron number, we have a(1) = hypericosahedron(7) = 7*((145*7^3)-(280*7^2)+(179*7)-38)/6 = 43435; and a(2) = hypericosahedron(hypericosahedron(7)) = hypericosahedron(43435) = 43435*((145*43435^3)-(280*43435^2)+(179*43435)-38)/6 = 86011544680330349395.
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REFERENCES
| Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969.
J. V. Post, "Iterated Polygonal Numbers", preprint.
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LINKS
| Hyun Kwang Kim, On Regular Polytope Numbers.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
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FORMULA
| a(0) = 2. Using the formula hypericosahedron(n) = n*((145*n^3)-(280*n^2)+(179*n)-38)/6 we have a(2) = hypericosahedron(2) = 2*((145*2^3)-(280*2^2)+(179*2)-38)/6 = 120. a(3) = hypericosahedron(hypericosahedron(2)) = hypericosahedron(120) = 4930988840. For k>0 we have the recurrence a(k+1) = hypericosahedron(a(k)) = a(k)*((145*a(k)^3)-(280*a(k)^2)+(179*a(k))-38)/6.
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EXAMPLE
| a(3) = 14287387711051307292599794275187472361080 because a(2) = 4930988840, hence a(3) = hypericosahedron(a(2)) = 4930988840*((145*4930988840^3)-(280*4930988840^2)+(179*4930988840)-38)/6 = 14287387711051307292599794275187472361080.
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CROSSREFS
| Cf. A092182, A099053, A099179, A000332, A006564.
Sequence in context: A024343 A100043 A181760 * A042799 A074490 A056638
Adjacent sequences: A100009 A100010 A100011 * A100013 A100014 A100015
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 17 2004
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