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A270516 Number of ordered ways to write n = x^3*(x+1) + y*(y+1)/2 + z*(3z+2), where x and y are nonnegative integers, and z is an integer.

%I A270516 #18 Mar 18 2016 15:48:58
%S A270516 1,2,2,3,2,2,3,2,4,2,3,4,1,3,1,2,3,3,3,2,2,3,4,3,5,3,4,2,4,4,3,5,2,5,
%T A270516 2,5,5,2,5,5,3,4,3,5,4,5,7,2,4,1,5,2,4,3,2,5,3,6,3,3,5,6,2,5,2,4,5,4,
%U A270516 8,3,4,5,1,5,3,1,4,3,5,4,5
%N A270516 Number of ordered ways to write n = x^3*(x+1) + y*(y+1)/2 + z*(3z+2), where x and y are nonnegative integers, and z is an integer.
%C A270516 Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and the only values of n > 1428 with a(n) = 1 are 2205, 2259, 3556, 4107, 4337, 5387, 9331, 16561, 22237, 27569, 63947, 78610.
%C A270516 (ii) Any natural number can be written as x*(x^3+2) + y*(y+1)/2 + z*(3z+1), where x and y are nonnegative integers, and z is an integer.
%C A270516 (iii) Every n = 0,1,2,... can be written as x*(x^3+x^2+6) + y*(y+1)/2 + z*(3z+2) (or x*(x^3+x^2+4x+1) + y*(y+1)/2 + z*(3z+1)), where x and y are nonnegative integers, and z is an integer.
%C A270516 See also A270533 for a similar conjecture.
%H A270516 Zhi-Wei Sun, <a href="/A270516/b270516.txt">Table of n, a(n) for n = 0..10000</a>
%H A270516 Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.
%H A270516 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2015.10.014">A result similar to Lagrange's theorem</a>, J. Number Theory 162(2016), 190-211.
%e A270516 a(72) = 1 since 72 = 2^3*3 + 5*6/2 + 3*(3*3+2).
%e A270516 a(75) = 1 since 75 = 0^3*1 + 4*5/2 + (-5)*(3*(-5)+2).
%e A270516 a(5387) = 1 since 5387 = 7^3*8 + 2*3/2 + (-30)*(3*(-30)+2).
%e A270516 a(9331) = 1 since 9331 = 8^3*9 + 2*3/2 + (-40)*(3*(-40)+2).
%e A270516 a(16561) = 1 since 16561 = 1^3*2 + 101*102/2 + (-62)*(3*(-62)+2).
%e A270516 a(22237) = 1 since 22237 = 6^3*7 + 104*105/2 + 71*(3*71+2).
%e A270516 a(27569) = 1 since 27569 = 2^3*3 + 49*50/2 + (-94)*(3*(-94)+2).
%e A270516 a(63947) = 1 since 63947 = 0^3*1 + 173*174/2 + (-128)*(3*(-128)+2).
%e A270516 a(78610) = 1 since 78610 = 16^3*17 + 52*53/2 + 50*(3*50+2).
%t A270516 OQ[x_]:=OQ[x]=IntegerQ[Sqrt[3x+1]]
%t A270516 Do[r=0;Do[If[OQ[n-y(y+1)/2-x^3*(x+1)],r=r+1],{y,0,(Sqrt[8n+1]-1)/2},{x,0,(n-y(y+1)/2)^(1/4)}];Print[n," ",r];Continue,{n,0,80}]
%Y A270516 Cf. A000217, A000578, A001082, A001318, A262813, A262815, A262816, A262941, A262944, A262945, A262954, A262955, A262956, A270469, A270488, A270533.
%K A270516 nonn
%O A270516 0,2
%A A270516 _Zhi-Wei Sun_, Mar 18 2016