

A242660


Nonnegative numbers of the form x^2+xy2y^2.


5



0, 1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34, 36, 37, 40, 43, 45, 46, 49, 52, 54, 55, 58, 61, 63, 64, 67, 70, 72, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 99, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 126, 127, 130, 133, 135, 136, 139, 142, 144, 145, 148, 151, 153, 154, 157, 160, 162, 163, 166, 169, 171, 172, 175, 178, 180, 181
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OFFSET

1,3


COMMENTS

Discriminant 9.
Are the positive entries the same as A056991?  R. J. Mathar, Jun 10 2014
We have x^2+xy2y^2 = (x+2y)(xy) which can be written as z(3x2z) by letting z=xy. All (x,z) pairs in the square 0<=x,z<=8 have values z(3x2z) == {0,1,4,7} (mod 9), which shows that all positive terms of this sequence have digital roots that define A056991: this sequence is a subsequence of A056991 (with 0 as a special case).  R. J. Mathar, Jun 12 2014


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..22223
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,1).


FORMULA

From Colin Barker, Oct 31 2016: (Start)
a(n) = a(n1)+a(n4)a(n5) for n>5.
G.f.: x^2*(1+2*x)*(1+x+x^2) / ((1x)^2*(1+x)*(1+x^2)).
(End)


MAPLE

# Maple Program fb, for indefinite binary quadratic forms
# f = ax^2+bxy+cy^2 with discriminant d = b^24ac = s^2 a perfect square.
# Looks for numbers 0 <= n <= M represented and also primes represented.
fb:=proc(a, b, c, M) local s, t1, t2, n, d, dp;
if not issqr(b^24*a*c) then error "disct not a square"; return; fi;
s:=sqrt(b^24*a*c); t1:={0}; t2:={};
for n from 1 to M do
for d in numtheory[divisors](4*a*n) do dp:=4*a*n/d;
if ((ddp) mod 2*s) = 0 and (((b+s)*dp(bs)*d) mod 4*a*s) = 0
then t1:={op(t1), n}; if isprime(n) then t2:={op(t2), n}; fi; break; fi;
od:
od:
[sort(convert(t1, list)), sort(convert(t2, list))];
end;
fb(1, 1, 2, 500);


MATHEMATICA

Select[Range[0, 1000], MatchQ[Mod[#, 9], Alternatives[0, 1, 4, 7]]&] (* JeanFrançois Alcover, Oct 31 2016 *)


PROG

(PARI) concat(0, Vec(x^2*(1+2*x)*(1+x+x^2)/((1x)^2*(1+x)*(1+x^2)) + O(x^100))) \\ Colin Barker, Oct 31 2016


CROSSREFS

Primes in this sequence = A002476.
Sequence in context: A266410 A010380 A056991 * A010389 A010415 A010442
Adjacent sequences: A242657 A242658 A242659 * A242661 A242662 A242663


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, May 31 2014, Jun 03 2014


STATUS

approved



