A334113 - OEIS

%I #14 May 30 2026 16:40:49

%S 23,44,54,63,117,138,149,162,180,188,243,251,261,270,287,294,398,401,

%T 458,512,611,657,684,693,734,797,842,863,914,932,936,945,987,1029,

%U 1047,1098,1323,1401,1449,1472,1484,1494,1574,1608,1637,1769,1792,1799,1823,1839,1902,1995

%N Positive numbers not of the form 4*x^4 + y*(y+1)/2 + z*(z+1)/2, where x,y,z are nonnegative integers.

%C Conjecture: The sequence only has 602 terms as listed in the b-file.

%C Our computation indicates that after the 602-th term 31737789 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 an even square and two triangular numbers.

%H Zhi-Wei Sun, <a href="/A334113/b334113.txt">Table of n, a(n) for n = 1..602</a>

%H Zhi-Wei Sun, <a href="https://doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.

%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];

%t tab={};Do[Do[If[TQ[n-4x^4-y(y+1)/2],Goto[aa]],{x,0,(n/4)^(1/4)},{y,0,(Sqrt[4(n-4x^4)+1]-1)/2}];tab=Append[tab,n];Label[aa],{n,0,2000}];Print[tab]

%Y Cf. A000217, A000583, A115160, A306227, A334086.

%K nonn,changed

%O 1,1

%A _Zhi-Wei Sun_, Apr 14 2020