|
%I #21 Jul 24 2023 10:28:35
%S 1,2,2,2,3,5,4,2,2,2,2,2,2,2,2,5,7,4,4,4,5,5,3,3,1,3,5,4,3,3,5,8,4,3,
%T 4,6,6,2,6,4,4,5,4,3,3,4,5,1,3,3,2,6,2,4,5,8,8,4,3,5,6,6,2,1,4,3,5,3,
%U 2,3,7,8,3,5,5,4,3,4,1,1,4
%N Number of ordered ways to write n as x^4 + y*(3y+1)/2 + z*(7z+1)/2, where x, y and z are integers with x nonnegative.
%C Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 24, 47, 63, 78, 79, 143, 153, 325, 494, 949, 1079, 3328, 4335, 5609, 7949, 7967, 8888, 9665.
%C Conjecture verified for n up to 10^11. - _Mauro Fiorentini_, Jul 24 2023
%H Zhi-Wei Sun, <a href="/A270566/b270566.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="https://www.sciengine.com/SCM/doi/10.1007/s11425-015-4994-4">On universal sums of polygonal numbers</a>, Sci. China Math. 58(2015), no. 7, 1367-1396.
%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(24) = 1 since 24 = 2^4 + (-2)*(3*(-2)+1)/2 + (-1)*(7*(-1)+1)/2.
%e a(78) = 1 since 78 = 1^4 + 7*(3*7+1)/2 + 0*(7*0+1)/2.
%e a(143) = 1 since 143 = 1^4 + 6*(3*6+1)/2 + (-5)*(7*(-5)+1)/2.
%e a(494) = 1 since 494 = 4^4 + (-7)*(3*(-7)+1)/2 + (-7)*(7*(-7)+1)/2.
%e a(949) = 1 since 949 = 4^4 + 0*(3*0+1)/2 + 14*(7*14+1)/2.
%e a(1079) = 1 since 1079 = 0^4 + 25*(3*25+1)/2 + 6*(7*6+1)/2.
%e a(3328) = 1 since 3328 = 0^4 + 38*(3*38+1)/2 + 18*(7*18+1)/2.
%e a(4335) = 1 since 4335 = 2^4 + 49*(3*49+1)/2 + 14*(7*14+1)/2.
%e a(5609) = 1 since 5609 = 0^4 + (-61)*(3*(-61)+1)/2 + 4*(7*4+1)/2.
%e a(7949) = 1 since 7949 = 3^4 + 43*(3*43+1)/2 + 38*(7*38+1)/2.
%e a(7967) = 1 since 7967 = 7^4 + (-61)*(3*(-61)+1)/2 + 2*(7*2+1)/2.
%e a(8888) = 1 since 8888 = 0^4 + (-77)*(3*(-77)+1)/2 + 3*(7*3+1)/2.
%e a(9665) = 1 since 9665 = 3^4 + 73*(3*73+1)/2 + 21*(7*21+1)/2.
%t (* From _Zhi-Wei Sun_, Start *)
%t pQ[n_] := pQ[n] = IntegerQ[Sqrt[24 n + 1]];
%t Do[r = 0; Do[If[pQ[n - x^4 - y (7 y + 1)/2], r = r + 1], {x, 0, n^(1/4)}, {y, -Floor[(Sqrt[56 (n - x^4) + 1] + 1)/14], (Sqrt[56 (n - x^4) + 1] - 1)/14}]; Print[n, " ", r]; Continue, {n, 0, 80}]
%t (* From _Zhi-Wei Sun_, End *)
%t A270566[n_] := Length@Solve[x >= 0 && n == x^4 + y*(3 y + 1)/2 + z*(7 z + 1)/2, {x, y, z}, Integers];
%t Array[A270566, 25, 0] (* _JungHwan Min_, Mar 19 2016 *)
%Y Cf. A001318, A000583, A262813, A262815, A262816, A262827, A262941, A262944, A262945, A262954, A262955, A262956, A270469, A270488, A270516, A270533, A270559.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Mar 19 2016
|
