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A351723 Numbers of the form x^2 + y^2 + z^2 + x*y*z with x,y,z nonnegative integers.

%I A351723 #26 Feb 27 2022 11:35:57
%S A351723 0,1,2,4,5,8,9,10,13,14,16,17,18,20,22,25,26,28,29,32,34,36,37,38,40,
%T A351723 41,44,45,49,50,52,53,54,58,61,62,64,65,68,70,72,73,74,76,77,80,81,82,
%U A351723 85,88,89,90,92,94,97,98,100,101,104,106,108,109,110,112,113,116,117,118,121,122,125,128,130,133,134,136
%N A351723 Numbers of the form x^2 + y^2 + z^2 + x*y*z with x,y,z nonnegative integers.
%C A351723 It is easy to see that no term is congruent to 3 modulo 4.
%C A351723 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.
%C A351723 Conjecture 2: Let S = {x^2 + y^2 + z^2 + x*y*z: x,y,z = 0,1,2,...}.
%C A351723 (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.
%C A351723 (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.
%C A351723 This has been verified for nonnegative integers up to 10^6.
%H A351723 Zhi-Wei Sun, <a href="/A351723/b351723.txt">Table of n, a(n) for n = 1..10000</a>
%H A351723 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/416344">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,...}?</a> Question 416344 at MathOverflow, Feb. 17, 2022.
%e A351723 a(3) = 2 with 2 = 1^2 + 1^2 + 0^2 + 1*1*0.
%t A351723 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]]]
%Y A351723 Cf. A000290, A000378, A000578, A351902.
%K A351723 nonn
%O A351723 1,3
%A A351723 _Zhi-Wei Sun_, Feb 17 2022