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 A267137 #44 Apr 05 2025 10:12:22
%S A267137 0,1,2,4,5,6,8,9,10,12,14,16,17,20,21,22,24,25,26,30,32,33,34,36,37,
%T A267137 40,41,42,44,46,49,50,52,54,56,57,58,60,64,65,66,69,70,72,74,76,80,81,
%U A267137 82,85,86,89,90,92,94,96,97,100,101,102,104,105,108,110,112,114,116
%N A267137 Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.
%C A267137 Inspired by relation between A003136 and A202822. See comment section of A202822.
%C A267137 Prime terms of this sequence are 2, 5, 17, 37, 41, 89, 97, 101, 137, 149, ...
%C A267137 Perfect power terms of this sequence are 1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 144, 169, ...
%C A267137 Obviously, A000290, A002378 and A045944 are subsequences.
%C A267137 The complement of this sequence is A322430. - _Kemoneilwe Thabo Moseki_, Dec 12 2019
%H A267137 Michael De Vlieger, <a href="/A267137/b267137.txt">Table of n, a(n) for n = 1..10583</a>
%H A267137 Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, <a href="https://arxiv.org/abs/2409.10359">Hilbert Space Fragmentation in the Chiral Luttinger Liquid</a>, arXiv:2409.10359 [cond-mat.str-el], 2024. See pp. 5, 8.
%H A267137 Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, <a href="https://doi.org/10.1103/PhysRevB.111.165105">Momentum-space modulated symmetries in the Luttinger liquid</a>, Phys. Rev. B (2025) Vol. 111, Art. No. 165105. See p. 4.
%F A267137 a(n) = (A202822(n) - 1) / 3.
%e A267137 1 is a term because (-1)^2 + (-1) + (-1)*(-1) + (-1) + (-1)^2 = 1.
%e A267137 4 is a term because 2^2 + 2 + 2*(-2) + (-2) + (-2)^2 = 4.
%e A267137 24 is a term because 2^2 + 2 + 2*3 + 3 + 3^2 = 24.
%t A267137 f[{i_, j_}] := (i^2 + i*j + j^2 + i + j); Union@ Map[f, Tuples[Range[-10, 10], 2] ] (* _Michael De Vlieger_, Sep 23 2024, after _Harvey P. Dale_ at A202822 *)
%o A267137 (PARI) x='x+O('x^500); p=eta(x)^3/eta(x^3); for(n=0, 499, if(polcoeff(p, n) != 0 && n%3==1, print1((n-1)/3, ", ")));
%o A267137 (PARI) is(n) = sumdiv( n, d, kronecker( -3, d));
%o A267137 for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));
%o A267137 (PARI) is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
%o A267137 for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));
%Y A267137 Cf. A000290, A002378, A003136, A045944, A202822.
%K A267137 nonn
%O A267137 1,3
%A A267137 _Altug Alkan_, Jan 10 2016