A020668 Numbers of the form x^2 + 4*y^2.

0, 1, 4, 5, 8, 9, 13, 16, 17, 20, 25, 29, 32, 36, 37, 40, 41, 45, 49, 52, 53, 61, 64, 65, 68, 72, 73, 80, 81, 85, 89, 97, 100, 101, 104, 109, 113, 116, 117, 121, 125, 128, 136, 137, 144, 145, 148, 149, 153, 157, 160, 164, 169, 173, 180, 181, 185, 193, 196, 197, 200, 205, 208

Offset: 1

Author

David W. Wilson

Keywords

• easy
• nonn

Comments

x^2 + 4y^2 has discriminant -16.

Numbers that can be expressed as both the sum of two squares and the difference of two squares; the intersection of sequences A001481 and A042965. - T. D. Noe, Feb 05 2003

A004531(n) is nonzero if and only if n is of the form x^2 + 4*y^2. - Michael Somos, Jan 05 2012

These are the sum of two squares that are congruent to 0 or 1 (mod 4), and thus that are also the difference of two squares. - Jean-Christophe Hervé, Oct 25 2015

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..7500
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A001481, A004531, A042965, A097269. For primes see A002144.

Programs

Magma

[n: n in [0..208] | NormEquation(4, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

Mathematica

Select[Range[0, 300], SquaresR[2, #] != 0 && Mod[#, 4] != 2&] (* Jean-François Alcover, May 13 2017 *)

PARI

for(n=0, 1e3, if(if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n]) != 0, print1(n, ", "))) \\ Altug Alkan, Oct 29 2015

Formula

Complement of A097269 in A001481. - Jean-Christophe Hervé, Oct 25 2015