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%I A282349 #6 Feb 12 2017 21:07:50
%S A282349 1,7,21,35,35,21,14,43,105,140,105,42,28,105,210,210,105,21,35,147,
%T A282349 252,245,175,105,77,154,315,455,420,210,63,147,441,630,420,105,7,147,
%U A282349 441,525,350,210,106,126,322,567,735,560,210,84,301,840,1050,630,210,49,315,875,980,630,245,35,245,707,1050,980,560,210,168
%N A282349 Expansion of (Sum_{k>=0} x^(k*(2*k^2+1)/3))^7.
%C A282349 Number of ways to write n as an ordered sum of 7 octahedral numbers (A005900).
%C A282349 Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers (a(n) > 0 for all n >= 0).
%H A282349 Ilya Gutkovskiy, <a href="/A282349/a282349.pdf">Extended graphical example</a>
%H A282349 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctahedralNumber.html">Octahedral Number</a>
%F A282349 G.f.: (Sum_{k>=0} x^(k*(2*k^2+1)/3))^7.
%e A282349 a(6) = 14 because we have:
%e A282349 [6, 0, 0, 0, 0, 0, 0]
%e A282349 [0, 6, 0, 0, 0, 0, 0]
%e A282349 [0, 0, 6, 0, 0, 0, 0]
%e A282349 [0, 0, 0, 6, 0, 0, 0]
%e A282349 [0, 0, 0, 0, 6, 0, 0]
%e A282349 [0, 0, 0, 0, 0, 6, 0]
%e A282349 [0, 0, 0, 0, 0, 0, 6]
%e A282349 [1, 1, 1, 1, 1, 1, 0]
%e A282349 [1, 1, 1, 1, 1, 0, 1]
%e A282349 [1, 1, 1, 1, 0, 1, 1]
%e A282349 [1, 1, 1, 0, 1, 1, 1]
%e A282349 [1, 1, 0, 1, 1, 1, 1]
%e A282349 [1, 0, 1, 1, 1, 1, 1]
%e A282349 [0, 1, 1, 1, 1, 1, 1]
%t A282349 nmax = 68; CoefficientList[Series[Sum[x^(k (2 k^2 + 1)/3), {k, 0, nmax}]^7, {x, 0, nmax}], x]
%Y A282349 Cf. A005900, A282172.
%K A282349 nonn
%O A282349 0,2
%A A282349 _Ilya Gutkovskiy_, Feb 12 2017

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