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%I #13 Jan 31 2019 05:14:34
%S 1,2,3,2,1,1,2,2,2,2,3,2,2,3,4,2,1,2,3,2,1,3,3,2,3,3,3,2,4,4,2,2,3,4,
%T 2,4,5,4,3,2,5,3,2,3,4,4,1,2,3,3,3,4,6,3,3,3,4,3,3,4,4,2,3,4,5,5,4,4,
%U 2,2,5,8,7,4,4,5,3,5,6,7,2,3,5,3,5,2,5,5,4,4,3,6,5,4,6,3,2,4,8,5,5
%N Number of ways to write n as x^3 + y^3 + pen(z) + pen(w), where x, y, z, w are nonnegative integers with x <= y and z <= w, and pen(k) denotes the pentagonal number k*(3*k-1)/2.
%C Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 4, 5, 16, 20, 46. Also, any nonnegative integer not equal to 16 can be written as x^6 + y^3 + pen(z) + pen(w) with x, y, z, w nonnegative integers.
%C We have verified a(n) > 0 for all n = 0..5*10^6.
%H Zhi-Wei Sun, <a href="/A306239/b306239.txt">Table of n, a(n) for n = 0..10000</a>
%e a(4) = 1 with 4 = 1^3 + 1^3 + pen(1) + pen(1).
%e a(5) = 1 with 5 = 0^3 + 0^3 + pen(0) + pen(2).
%e a(16) = 1 with 16 = 2^3 + 2^3 + pen(0) + pen(0).
%e a(20) = 1 with 20 = 0^3 + 2^3 + pen(0) + pen(3).
%e a(46) = 1 with 46 = 1^3 + 1^3 + pen(4) + pen(4).
%t PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0);
%t tab={};Do[r=0;Do[If[PenQ[n-x^3-y^3-z(3z-1)/2],r=r+1],{x,0,(n/2)^(1/3)},{y,x,(n-x^3)^(1/3)},{z,0,(Sqrt[12(n-x^3-y^3)+1]+1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y Cf. A000326, A000578, A306225, A306227.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Jan 31 2019
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