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%I #20 Apr 15 2020 15:09:16
%S 19,82,109,118,145,149,271,280,296,349,350,371,392,454,491,643,670,
%T 692,754,755,923,937,986,989,1021,1031,1150,1189,1210,1294,1346,1372,
%U 1610,1682,1699,1720,1819,1913,2050,2065,2141,2227,2479,2524,2753,2996,3184,3451,3590,3805,3968,4129,4139,4199,4261,4706
%N Positive numbers not of the form 2*x^4 + y*(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers
%C Conjecture: The sequence has totally 216 terms as listed in the b-file.
%C As none of the 216 terms in the b-file is divisible by 3, the conjecture implies that for each nonnegative integer n we can write 3*n as 2*x^4 + y*(y+1)/2 + z*(z+1)/2 and hence 12*n+1 = 8*x^4 + (y+z+1)^2 + (y-z)^2, where x,y,z are integers.
%C Our computation indicates that after the 216-th term 4592329 there are no other terms below 10^8.
%C It is known that each n = 0,1,2,... can be written as the sum of two triangular numbers and twice a square.
%C a(217) > 10^9, if it exists. - _Giovanni Resta_, Apr 14 2020
%H Zhi-Wei Sun, <a href="/A334086/b334086.txt">Table of n, a(n) for n = 1..216</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120. (Cf. Conjecture 1.4(ii).)
%e a(1) = 19 since 19 is the first nonnegative integer which cannot be written as the sum of two triangular numbers and twice a fourth power.
%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
%t tab={};Do[Do[If[TQ[n-2x^4-y(y+1)/2],Goto[aa]],{x,0,(n/2)^(1/4)},{y,0,(Sqrt[4(n-2x^4)+1]-1)/2}];tab=Append[tab,n];Label[aa],{n,0,5000}];Print[tab]
%Y Cf. A000217, A000583, A115160, A290491, A306227, A334113.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Apr 14 2020
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