OFFSET
1,1
COMMENTS
Conjecture: The sequence has totally 216 terms as listed in the b-file.
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.
Our computation indicates that after the 216-th term 4592329 there are no other terms below 10^8.
It is known that each n = 0,1,2,... can be written as the sum of two triangular numbers and twice a square.
a(217) > 10^9, if it exists. - Giovanni Resta, Apr 14 2020
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..216
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120. (Cf. Conjecture 1.4(ii).)
EXAMPLE
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.
MATHEMATICA
TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 14 2020
STATUS
approved