login
A267137
Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.
6
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
1,3
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
LINKS
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.
FORMULA
a(n) = (A202822(n) - 1) / 3.
EXAMPLE
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.
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 *)
PROG
(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, ", ")));
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Jan 10 2016
STATUS
approved