A334113 Positive numbers not of the form 4*x^4 + y*(y+1)/2 + z*(z+1)/2, where x,y,z are nonnegative integers.

23, 44, 54, 63, 117, 138, 149, 162, 180, 188, 243, 251, 261, 270, 287, 294, 398, 401, 458, 512, 611, 657, 684, 693, 734, 797, 842, 863, 914, 932, 936, 945, 987, 1029, 1047, 1098, 1323, 1401, 1449, 1472, 1484, 1494, 1574, 1608, 1637, 1769, 1792, 1799, 1823, 1839, 1902, 1995

Offset: 1

Author

Zhi-Wei Sun, Apr 14 2020

Keywords

• nonn

Comments

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

Our computation indicates that after the 602-th term 31737789 there are no other terms below 10^8.

It is known that each n = 0,1,2,... can be written as the sum of an even square and two triangular numbers.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..602
• Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Crossrefs

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

Programs

Mathematica

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