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a(0) = 2 and a(n) = f(a(n-1)) where f(n) = n^2*(3*n^2-4*n+2).
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%I #29 Aug 30 2024 10:19:11

%S 2,24,941184,2354066797535483525627904

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

%C 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.

%C The next term has 98 digits.

%C 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.

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

%H Michel Marcus, <a href="/A100010/b100010.txt">Table of n, a(n) for n = 0..5</a>

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2003), 65-75.

%H Jonathan Vos Post, <a href="https://web.archive.org/web/20100701204436/http://magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.

%F 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)).

%e a(0) = 2 is the seed for this instance of the more general recurrence;

%e a(1) = 24-cell(2) = 2^2*(3*2^2-4*2+2) = 24;

%e a(2) = 24-cell(24-cell(2)) = 24-cell(24) = 24^2*(3*24^2-4*24+2) = 941184.

%o (PARI) f(n) = n^2*(3*n^2-4*n+2); \\ A092181

%o a(n) = if (n==0, 2, f(a(n-1))); \\ _Michel Marcus_, Dec 14 2015

%Y Cf. A092181, A007501, A099179, A000332.

%K easy,nonn

%O 0,1

%A _Jonathan Vos Post_, Nov 16 2004

%E New name from _Joerg Arndt_, Feb 23 2022