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 A263998 #5 Oct 31 2015 14:01:30
%S A263998 1,1,3,3,3,3,4,1,4,4,3,7,2,4,5,2,3,4,7,3,7,5,4,5,5,3,5,8,3,8,3,4,6,5,
%T A263998 4,5,10,2,11,4,2,6,3,6,3,7,5,5,3,3,6,5,6,8,7,3,9,5,4,9,5,4,4,8,4,5,8,
%U A263998 2,11,5,5,9,5,6,8,6,5,10,8,3,4,13,4,10,7,4,12,6,7,4,10,6,7,6,4,9,5,5,8,11
%N A263998 Number of ordered ways to write n as x^2 + 2*y^2 + p*(p+d)/2, where x and y are nonnegative integers, d is 1 or -1, and p is prime.
%C A263998 Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 8.
%C A263998 This is similar to the conjecture in A262785.
%H A263998 Zhi-Wei Sun, <a href="/A263998/b263998.txt">Table of n, a(n) for n = 1..10000</a>
%H A263998 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6f4ac126.1510">Some mysterious representations of integers</a>, a message to Number Theory Mailing List, Oct. 25, 2015.
%e A263998 a(1) = 1 since 1 = 0^2 + 2*0^2 + 2*(2-1)/2 with 2 prime.
%e A263998 a(2) = 1 since 2 = 1^2 + 2*0^2 + 2*(2-1)/2 with 2 prime.
%e A263998 a(8) = 1 since 8 = 0^2 + 2*1^2 + 3*(3+1)/2 with 3 prime.
%t A263998 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A263998 f[d_, n_]:=f[d,n]=Prime[n](Prime[n]+(-1)^d)/2
%t A263998 Do[r=0; Do[If[SQ[n-f[d, k]-2x^2], r=r+1], {d, 0, 1}, {k, 1, PrimePi[(Sqrt[8n+1]-(-1)^d)/2]}, {x, 0, Sqrt[(n-f[d, k])/2]}]; Print[n, " ", r]; Continue, {n, 1, 100}]
%Y A263998 Cf. A000040, A000217, A000290, A262311, A262785, A263992.
%K A263998 nonn
%O A263998 1,3
%A A263998 _Zhi-Wei Sun_, Oct 31 2015