OFFSET
1,3
COMMENTS
It is easy to see that no term is congruent to 3 modulo 4.
Conjecture 1: a(n) < 2*n for all n > 0, and a(n)/n has a limit as n tends to the infinity. Also, a(n) <= a(n-1) + a(n-2) for all n > 4.
Conjecture 2: Let S = {x^2 + y^2 + z^2 + x*y*z: x,y,z = 0,1,2,...}.
(i) 7 and 487 are the only nonnegative integers which cannot be written as w^2 + s, where w is a nonnegative integer and s is an element of S. Also, 7, 87 and 267 are the only nonnegative integers which cannot be written as w^3 + s, where w is a nonnegative integer and s is an element of S.
(ii) Let k be 2 or 3. Then each nonnegative integer not congruent to 3 modulo 4 can be written as 4*w^k + s, where w is a nonnegative integer and s is an element of S.
This has been verified for nonnegative integers up to 10^6.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Can each natural number be represented by 2*w^2+x^2+y^2+z^2+xyz with x,y,z in {0,1,2,...}? Question 416344 at MathOverflow, Feb. 17, 2022.
EXAMPLE
a(3) = 2 with 2 = 1^2 + 1^2 + 0^2 + 1*1*0.
MATHEMATICA
tab={}; Do[n=x^2+y^2+z^2+x*y*z; If[n<=140, tab=Append[tab, n]], {x, 0, 20}, {y, 0, x}, {z, 0, y}]; Print[Sort[DeleteDuplicates[tab]]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 17 2022
STATUS
approved