A267137 Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

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, 40, 41, 42, 44, 46, 49, 50, 52, 54, 56, 57, 58, 60, 64, 65, 66, 69, 70, 72, 74, 76, 80, 81, 82, 85, 86, 89, 90, 92, 94, 96, 97, 100, 101, 102, 104, 105, 108, 110, 112, 114, 116

Offset: 1

Author

Altug Alkan, Jan 10 2016

Keywords

• nonn

Comments

Inspired by relation between A003136 and A202822. See comment section of A202822.

Prime terms of this sequence are 2, 5, 17, 37, 41, 89, 97, 101, 137, 149, ...

Perfect power terms of this sequence are 1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 144, 169, ...

Obviously, A000290, A002378 and A045944 are subsequences.

The complement of this sequence is A322430. - Kemoneilwe Thabo Moseki, Dec 12 2019

Examples

			1 is a term because (-1)^2 + (-1) + (-1)*(-1) + (-1) + (-1)^2 = 1.
4 is a term because 2^2 + 2 + 2*(-2) + (-2) + (-2)^2 = 4.
24 is a term because 2^2 + 2 + 2*3 + 3 + 3^2 = 24.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10583
• Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Hilbert Space Fragmentation in the Chiral Luttinger Liquid, arXiv:2409.10359 [cond-mat.str-el], 2024. See pp. 5, 8.
• Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Momentum-space modulated symmetries in the Luttinger liquid, Phys. Rev. B (2025) Vol. 111, Art. No. 165105. See p. 4.

Crossrefs

Cf. A000290, A002378, A003136, A045944, A202822.

Programs

Mathematica

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 *)

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, ", ")));

PARI

is(n) = sumdiv( n, d, kronecker( -3, d));
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));

PARI

is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));

Formula

a(n) = (A202822(n) - 1) / 3.