A100010 a(0) = 2 and a(n) = f(a(n-1)) where f(n) = n^2*(3*n^2-4*n+2).

2, 24, 941184, 2354066797535483525627904

Offset: 0

Author

Jonathan Vos Post, Nov 16 2004

Keywords

• easy
• nonn

Comments

Previous name was: Iterated hyperdiamond numbers, starting with 24-cell(2) = 24. Hyperdiamond numbers, figurate numbers based on the 4-dimensional 24-cell, have the formula 24-cell(n) = n^2*(3*n^2-4*n+2). This sequence is the hyperdiamond number of the hyperdiamond number of ... of 2.

The next term has 98 digits.

This need not start at 24-cell(2) = 24. For example, starting at a(0) = 3, which is not a hyperdiamond number, we have a(1) = 24-cell(3) = 3^2*((3*3^2)-(4*3)+2) = 153; and a(2) = 24-cell(24-cell(3)) = 24-cell(153) = 153^2*((3*153^2)-(4*153)+2) = 1629664353; and a(3) = 24-cell(24-cell(24-cell(3))) = 24-cell(1629664353) = 21159914972910583843562449776792301953.

Examples

			a(0) = 2 is the seed for this instance of the more general recurrence;
a(1) = 24-cell(2) = 2^2*(3*2^2-4*2+2) = 24;
a(2) = 24-cell(24-cell(2)) = 24-cell(24) = 24^2*(3*24^2-4*24+2) = 941184.
		

References

• H. S. M. Coxeter, Regular Polytopes, 3rd ed. New York: Dover, 1973.

Links

• Michel Marcus, Table of n, a(n) for n = 0..5
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
• Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

Crossrefs

Cf. A092181, A007501, A099179, A000332.

Programs

PARI

f(n) = n^2*(3*n^2-4*n+2); \\ A092181
a(n) = if (n==0, 2, f(a(n-1))); \\ Michel Marcus, Dec 14 2015

Formula

a(0) = 2; hyperdiamond numbers, figurate numbers based on the 4-dimensional 24-cell, have the formula 24-cell(n) = n^2*(3*n^2-4*n+2). a(1) = 24-cell(2) = 24. a(2) = 24-cell(24-cell(2)) = 941184. For k>1, a(k+1) = 24-cell(a(k)).

Extensions

New name from Joerg Arndt, Feb 23 2022