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%I #12 Oct 01 2016 04:41:27
%S 1,1,2,2,2,1,3,1,2,2,2,3,3,3,2,2,4,2,3,1,3,3,3,3,2,2,3,2,2,2,4,6,3,3,
%T 3,1,5,3,4,4,3,4,3,2,3,3,6,2,5,2,2,5,3,3,1,4,4,4,5,3,3,5,1,1,2,3,7,4,
%U 5,4,3,3,6,2,5,4,6,2,5,4,3
%N Number of ordered ways to write n = x^2 + y*(y+1) + z*(z^2+1), where x, y and z are nonnegative integers.
%C Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 5, 7, 19, 35, 54, 62, 63, 197, 285, 339, 479, 505, 917. Moreover, any integer n > 2 can be written as x^2 + y*(y+1) + z*(z^2+1), where x is a positive integer, and y and z are nonnegative integers.
%C We also guess that each n = 0,1,2,... can be expressed as x*(x+1)/2 + P(y,z) with x, y and z nonnegative integers, where P(y,z) is any of the polynomials y(y+1) + z^2*(z+1), y^2 + z*(z^2+2), y^2 + z*(z^2+7), y^2 + z*(z^2+z+2), y^2 + z*(z^2+2z+3), y^2 + z*(2z^2+z+1).
%C It is known that every n = 0,1,2,... can be written as x^2 + y*(y+1) + z*(z+1), where x, y and z are nonnegative integers.
%H Zhi-Wei Sun, <a href="/A270488/b270488.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.07.024">On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d)</a>, J. Number Theory 171(2017), 275-283.
%e a(35) = 1 since 35 = 5^2 + 0*1 + 2*(2^2+1).
%e a(54) = 1 since 54 = 2^2 + 4*5 + 3*(3^2+1).
%e a(62) = 1 since 62 = 2^2 + 7*8 + 1*(1^2+1).
%e a(63) = 1 since 63 = 7^2 + 3*4 + 1*(1^2+1).
%e a(197) = 1 since 197 = 5^2 + 6*7 + 5*(5^2+1).
%e a(285) = 1 since 285 = 15^2 + 5*6 + 3*(3^2+1).
%e a(339) = 1 since 339 = 17^2 + 4*5 + 3*(3^2+1).
%e a(479) = 1 since 479 = 7^2 + 20*21 + 2*(2^2+1).
%e a(505) = 1 since 505 = 13^2 + 17*18 + 3*(3^2+1).
%e a(917) = 1 since 917 = 15^2 + 18*19 + 7*(7^2+1).
%t SQ[x_]:=SQ[x]=IntegerQ[Sqrt[x]]
%t Do[r=0;Do[If[SQ[n-y(y+1)-z(z^2+1)],r=r+1],{y,0,(Sqrt[4n+1]-1)/2},{z,0,(n-y(y+1))^(1/3)}];Print[n," ",r];Continue,{n,0,80}]
%Y Cf. A000217, A000290, A000578, A002378, A002522, A262813, A262815, A262816, A270469.
%K nonn
%O 0,3
%A _Zhi-Wei Sun_, Mar 17 2016
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