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%I #28 Nov 20 2023 23:03:42
%S 126,182,217,237,247,251,266,301,321,331,335,357,377,386,387,391,412,
%T 421,422,426,441,442,446,451,455,456,477,497,507,511,532,542,546,551,
%U 561,562,566,576,581,586,591,595,606,616,620,626,630,642,646,650
%N Sums of 5 distinct positive pentatope numbers (A000332).
%C Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.
%D Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.
%H Robert Israel, <a href="/A104395/b104395.txt">Table of n, a(n) for n = 1..10000</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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentatopeNumber.html">Pentatope Number</a>.
%F a(n) = Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).
%p N:= 1000: # for terms <= N
%p ptop:= n -> n*(n+1)*(n+2)*(n+3)/24:
%p P:= 1:
%p for i from 1 while ptop(i) < N do
%p P:= P * (1 + x*y^ptop(i))
%p od:
%p sort(map(degree,convert(convert(series(coeff(P,x,5),y,N+1),polynom),list)));
%p # _Robert Israel_, Nov 20 2023
%Y Cf. A000332, A100009, A102857, A104392, A104393, A104394.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Mar 05 2005
%E Extended by _Ray Chandler_, Mar 05 2005
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