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%I #15 Apr 11 2020 06:26:54
%S 1,2,2,2,2,2,2,2,3,3,2,1,3,5,4,3,2,1,1,2,4,4,3,3,2,2,3,4,3,2,3,4,5,6,
%T 4,2,2,2,2,3,5,5,4,4,4,4,2,1,3,4,5,5,3,2,2,2,3,4,4,5,4,2,4,6,5,2,2,3,
%U 4,6,6,4,4,5,3,3,6,6,4,3,3,3,3,3,5,7,6,5,3,3,4,3,5,6,4,3,4,4,3,5,6
%N Number of ways to write n as x^9 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2 and z <= w.
%C Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 11, 17, 18, 47, 108, 109, 234, 359. Also, any nonnegative integer can be written as x^6 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2.
%C We have verified a(n) > 0 for all n = 0..2*10^7.
%H Zhi-Wei Sun, <a href="/A306240/b306240.txt">Table of n, a(n) for n = 0..10000</a>
%e a(11) = 1 with 11 = 1^9 + 2^3 + 0*1 + 1*2.
%e a(18) = 1 with 18 = 0^9 + 0^3 + 2*3 + 3*4.
%e a(109) = 1 with 109 = 1^9 + 4^3 + 1*2 + 6*7.
%e a(234) = 1 with 234 = 0^9 + 6^3 + 2*3 + 3*4.
%e a(359) = 1 with 359 = 1^9 + 2^3 + 10*11 + 15*16.
%e a(1978) = 3 with 1978 = 2^9 + 2^3 + 26*27 + 27*28 = 2^9 + 6^3 + 19*20 + 29*30 = 2^9 + 6^3 + 24*25 + 25*26.
%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];
%t tab={};Do[r=0;Do[If[TQ[n-x^9-y^3-z(z+1)],r=r+1],{x,0,Min[2,n^(1/9)]},{y,0,(n-x^9)^(1/3)},{z,0,(Sqrt[2(n-x^9-y^3)+1]-1)/2}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y Cf. A000578, A001014, A001017, A002378, A262813, A262815, A270488, A306225, A306227, A306239.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Jan 31 2019
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