This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A271026 #10 Mar 29 2016 04:44:17
%S A271026 1,4,7,7,4,2,3,4,5,6,5,3,2,4,5,4,6,7,5,3,2,3,4,6,8,5,3,5,7,8,6,5,5,3,
%T A271026 3,5,6,4,2,4,5,4,5,7,6,3,2,1,2,4,5,5,5,5,3,2,2,3,5,6,4,1,1,2,3,6,7,6,
%U A271026 5,4,4,5,5,3,2,2,2,3,7,9,6
%N A271026 Number of ordered ways to write n as x^7 + y^4 + z^3 + w*(3w+1)/2, where x, y, z are nonnegative integers, and w is an integer.
%C A271026 Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 47, 61, 62, 112, 175, 448, 573, 714, 1073, 1175, 1839, 2167, 8043, 13844.
%C A271026 (ii) Any natural number can be written as 3*x^6 + y^4 + z^3 + w*(3w+1)/2, where x, y, z are nonnegative integers and w is an integer.
%C A271026 (iii) For every a = 3, 4, 5, 9, 12, any natural number can be written as a*x^5 + y^4 + z^3 + w*(3w+1)/2, where x, y, z are nonnegative integers and w is an integer. Also, any natural number can be written as x^5 + 2*y^4 + 2*z^3 + w*(3w+1)/2 (or 3*x^5 + 2*y^4 + z^3 + w*(3w+1)/2), where x, y, z are nonnegative integers and w is an integer.
%C A271026 We have verified that a(n) > 0 for n up to 2*10^6.
%C A271026 See also A266968 for a related conjecture.
%H A271026 Zhi-Wei Sun, <a href="/A271026/b271026.txt">Table of n, a(n) for n = 0..10000</a>
%H A271026 Z.-W. 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 A271026 Z.-W. Sun, <a href="http://math.scichina.com:8081/sciAe/EN/abstract/abstract517007.shtml">On universal sums of polygonal numbers</a>, Sci. China Math. 58(2015), 1367-1396.
%e A271026 a(47) = 1 since 47 = 1^7 + 2^4 + 2^3 + (-4)*(3*(-4)+1)/2.
%e A271026 a(61) = 1 since 61 = 1^7 + 1^4 + 2^3 + (-6)*(3*(-6)+1)/2.
%e A271026 a(62) = 1 since 62 = 0^7 + 0^4 + 3^3 + (-5)*(3*(-5)+1)/2.
%e A271026 a(112) = 1 since 112 = 1^7 + 3^4 + 2^3 + (-4)*(3*(-4)+1)/2.
%e A271026 a(175) = 1 since 175 = 1^7 + 3^4 + 1^3 + (-8)*(3*(-8)+1)/2.
%e A271026 a(448) = 1 since 448 = 2^7 + 4^4 + 4^3 + 0*(3*0+1)/2.
%e A271026 a(573) = 1 since 573 = 1^7 + 4^4 + 6^3 + 8*(3*8+1)/2.
%e A271026 a(714) = 1 since 714 = 2^7 + 4^4 + 0^3 + (-15)*(3*(-15)+1)/2.
%e A271026 a(1073) = 1 since 1073 = 0^7 + 2^4 + 10^3 + 6*(3*6+1)/2.
%e A271026 a(1175) = 1 since 1175 = 0^7 + 5^4 + 5^3 + (-17)*(3*(-17)+1)/2.
%e A271026 a(1839) = 1 since 1839 = 1^7 + 4^4 + 5^3 + 31*(3*31+1)/2.
%e A271026 a(2167) = 1 since 2167 = 1^7 + 5^4 + 11^3 + (-12)*(3*(-12)+1)/2.
%e A271026 a(8043) = 1 since 8043 = 1^7 + 2^4 + 20^3 + 4*(3*4+1)/2.
%e A271026 a(13844) = 1 since 13844 = 3^7 + 2^4 + 21^3 + (-40)*(3*(-40)+1)/2.
%t A271026 pQ[n_]:=pQ[n]=IntegerQ[Sqrt[24n+1]]
%t A271026 Do[r=0;Do[If[pQ[n-x^7-y^4-z^3],r=r+1],{x,0,n^(1/7)},{y,0,(n-x^7)^(1/4)},{z,0,(n-x^7-y^4)^(1/3)}];Print[n," ",r];Continue,{n,0,80}]
%Y A271026 Cf. A000326, A000578, A000583, A000584, A001015, A001318, A262813, A262815, A262816, A262827, A266968, A270469, A270488, A270516, A270533, A270559, A270566, A270920.
%K A271026 nonn
%O A271026 0,2
%A A271026 _Zhi-Wei Sun_, Mar 29 2016