A084917 -id:A084917 - cp's OEIS frontend

A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).

2, 3, 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008

nonn

Discriminant = 12. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.

This is exactly {2} U A068231, primes congruent to 11 (mod 12). This is because the orders of imaginary quadratic fields with discriminant 12 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - Jianing Song, Jun 22 2025

			a(3) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Robert Israel, Table of n, a(n) for n = 1..6514
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).
Essentially the same as A068231 and A141187.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. A084917.

N:= 2000:
S:= NULL:
for xx from 1 to floor(2*sqrt(N/3)) do
  for yy from ceil(sqrt(max(1,3*xx^2-N))) to floor(sqrt(3)*xx) do
     S:= S, 3*xx^2-yy^2;
od od:
sort(convert(select(isprime,{S}),list)); # Robert Israel, Jul 20 2020

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* Jean-François Alcover, Oct 25 2016, updated Oct 29 2016 *)

More terms from Colin Barker, Apr 05 2015

A142463 a(n) = 2n^2 + 2n - 1.

Original entry on oeis.org

-1, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511, 4703, 4899, 5099

Offset: 0

Roger L. Bagula, Sep 19 2008

Essentially the same as A132209.
From Vincenzo Librandi, Nov 25 2010: (Start)
Numbers k such that 2*k + 3 is a square.
First diagonal of A144562. (End)
The terms a(n) give the values for c of indefinite binary quadratic forms [a, b, c] = [2, 4n+2, a(n)] of discriminant D = 12, where a and c can be switched. The positive numbers represented by these forms are given in A084917. - Klaus Purath, Aug 31 2023

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Leo Tavares, Illustration: Hexagonic Diamonds.
Leo Tavares, Illustration: Hexagonic Rectangles.
Leo Tavares, Illustration: Hexagonic Crosses.
Leo Tavares, Illustration: Hexagonic Columns.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A005408, A132209, A144562, A188653.

Cf. A005563, A016945, A001105, A000290, A004767, A046092, A002378, A028387.

Magma
```
[2*n^2+2*n-1: n in [0..100]];
```

A142463:= n-> 2*n^2 +2*n -1; seq(A142463(n), n=0..50); # G. C. Greubel, Mar 01 2021

Array[ -#*(2-#*2)-1&,5!,1] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
Table[2n^2+2n-1,{n,0,50}] (* Harvey P. Dale, Feb 29 2024 *)

a(n)=2*n^2+2*n-1 \\ Charles R Greathouse IV, Sep 24 2015

[2*n^2 +2*n -1 for n in (0..50)] # G. C. Greubel, Mar 01 2021

a(n) = a(n-1) + 4*n.
From Paul Barry, Nov 03 2009: (Start)
G.f.: (1 - 6*x + x^2)/(1-x)^3.
a(n) = 4*C(n+1,2) - 1. (End)
a(n) = -A188653(2*n+1). - Reinhard Zumkeller, Apr 13 2011
a(n) = 3*( Sum_{k=1..n} k^5 )/( Sum_{k=1..n} k^3 ), n > 0. - Gary Detlefs, Oct 18 2011
a(n) = (A005408(n)^2 - 3)/2. - Zhandos Mambetaliyev, Feb 11 2017
E.g.f.: (-1 + 4*x + 2*x^2)*exp(x). - G. C. Greubel, Mar 01 2021
From Leo Tavares, Nov 22 2021: (Start)
a(n) = 2*A005563(n) - A005408(n). See Hexagonic Diamonds illustration.
a(n) = A016945(n-1) + A001105(n-1). See Hexagonic Rectangles illustration.
a(n) = A004767(n-1) + A046092(n-1). See Hexagonic Crosses illustration.
a(n) = A002378(n) + A028387(n-1). See Hexagonic Columns illustration. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Dec 03 2021
Sum_{n>=0} 1/a(n) = tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022

Edited by the Associate Editors of the OEIS, Sep 02 2009

A237606 Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

6, 11, 14, 15, 24, 35, 44, 51, 54, 56, 59, 60, 71, 86, 96, 99, 110, 119, 126, 131, 134, 135, 140, 150, 159, 176, 179, 191, 204, 206, 215, 216, 224, 231, 236, 239, 240, 251, 254, 275, 284, 294, 311, 315, 326, 335, 339, 344, 350, 359, 366, 371, 374, 375, 384

Offset: 1

Colin Barker, Feb 10 2014

nonn

From Klaus Purath, Feb 17 2024: (Start)
Positive numbers of the form 15x^2 - y^2. The reduced form is -x^2 + 6xy + 6y^2.
Even powers of terms as well as products of an even number of terms belong to A243188. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (au^2 - cv^2)(ax^2 - cy^2) = (aux + cvy)^2 - ac(uy + vx)^2 and (au^2 + cv^2)(ax^2 + cy^2) = (aux - cvy)^2 + ac(uy + vx)^2 for all a, c, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (as^2 - ct^2)(au^2 - cv^2)(ax^2 - cy^2) = a[s(aux + cvy) + ct(uy + vx)]^2 - c[as(uy + vx) + t(aux + cvy)]^2 and (as^2 + ct^2)(au^2 + cv^2)(ax^2 + cy^2) = a[s(aux - cvy) - ct(uy + vx)]^2 + c[as(uy + vx) + t(aux - cvy)]^2 for all a, c, s, t, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
If we denote any term of this sequence by B and correspondingly of A243189 by C and of A243190 by D, then B*C = D, C*D = B and B*D = C. This can be proved by the following identities, where the sequence (B) is represented by [kn, 0, -1], (C) by [n, 0, -k] and (D) by [k, 0, -n].
Proof of B*C = D: (knu^2 - v^2)(nx^2 - ky^2) =  k(nux + vy)^2 - n(kuy + vx)^2 for k, n, u, v, x, y in R.
Proof of C*D = B: (nu^2 - kv^2)(kx^2 - ny^2) = kn(ux + vy)^2 - (nuy + kvx)^2 for k, n, u, v, x, y in R.
Proof of B*D = C: (knu^2 - v^2)(kx^2 - ny^2) = n(kux + vy)^2 - k(nuy + vx)^2 for k, n, u, v, x, y in R. This can be verified by expanding both sides of the equations.
Generalization (conjecture): If there are three sequences of a given positive discriminant that are represented by the forms [a1, b1, c1], [a2, b2, c2] and [a1*a2, b3, c3] for a1, a2 != 1, then the BCD rules apply to these sequences. (End)

			6 is in the sequence because x^2 - 8xy + y^2 + 6 = 0 has integer solutions, for example (x, y) = (1, 7).

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A070997 (k = 6), A199336 (k = 14), A001091 (k = 15), A077248 (k = 35).
Cf. A031363, A084917, A237351, A237599, A237609, A237610.
For primes see A141302.
Cf. A378710, A378711 (subsequence of properly represented numbers and fundamental solutions).

A237351 Positive integers k such that x^2 - 5xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

3, 5, 12, 17, 20, 21, 27, 35, 41, 45, 47, 48, 59, 68, 75, 80, 83, 84, 89, 101, 108, 111, 119, 125, 129, 131, 140, 147, 153, 164, 167, 173, 180, 185, 188, 189, 192, 201, 215, 227, 236, 237, 243, 245, 251, 255, 257, 269, 272, 287, 293, 300, 311, 315, 320, 327

Offset: 1

Colin Barker, Feb 06 2014

nonn

See comments on method used in A084917.
The equivalent sequence for x^2 - 3xy + y^2 + k = 0 is A031363.
The equivalent sequence for x^2 - 4xy + y^2 + k = 0 is A084917.
Positive numbers of the form 3x^2 - 7y^2. - Jon E. Schoenfield, Jun 03 2022

			12 is in the sequence because x^2 - 5xy + y^2 + 12 = 0 has integer solutions, for example, (x, y) = (2, 8).

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A004253 (k = 3), A237254 (k = 5), A237255 (k = 17).
Cf. A031363, A084917.
For primes see A141160.

Select[Range[350],Length[FindInstance[x^2-5x y+y^2+#==0,{x,y},Integers]]>0&] (* Harvey P. Dale, Apr 23 2023 *)

A237599 Positive integers k such that x^2 - 6xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

4, 7, 8, 16, 23, 28, 31, 32, 36, 47, 56, 63, 64, 68, 71, 72, 79, 92, 100, 103, 112, 119, 124, 127, 128, 136, 144, 151, 164, 167, 175, 184, 188, 191, 196, 199, 200, 207, 223, 224, 239, 248, 252, 256, 263, 271, 272, 279, 284, 287, 288, 292, 311, 316, 324, 328

Offset: 1

Colin Barker, Feb 10 2014

nonn

Nonnegative numbers of the form 8x^2 - y^2. - Jon E. Schoenfield, Jun 03 2022

			4 is in the sequence because x^2 - 6xy + y^2 + 4 = 0 has integer solutions, for example (x, y) = (1, 5).

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A001653 (k = 4), A006452 (k = 7), A001541 (k = 8), A075870 (k = 16), A156066 (k = 23), A217975 (k = 28), A003499 (k = 32), A075841 (k = 36), A077443 (k = 56).
Cf. A031363, A084917, A237351, A237606, A237609, A237610.
For primes see A007522 and A141175.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

A237609 Positive integers k such that x^2 - 9xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

7, 13, 17, 19, 28, 41, 52, 61, 63, 68, 73, 76, 77, 83, 101, 112, 117, 131, 139, 143, 153, 161, 164, 167, 171, 173, 175, 187, 208, 209, 227, 241, 244, 252, 259, 271, 272, 283, 292, 293, 299, 304, 307, 308, 325, 332, 343, 349, 369, 371, 391, 404, 409, 425, 437

Offset: 1

Colin Barker, Feb 10 2014

nonn

			7 is in the sequence because x^2 - 9xy + y^2 + 7 = 0 has integer solutions, for example (x, y) = (1, 8).

Cf. A070998 (k = 7).

Cf. A031363, A084917, A237351, A237599, A237606, A237610.

is(n)=bnfisintnorm(bnfinit(x^2-9*x+1),-n) \\ Ralf Stephan, Feb 11 2014

A237610 Positive integers k such that x^2 - 10xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

8, 15, 20, 23, 24, 32, 47, 60, 71, 72, 80, 87, 92, 95, 96, 116, 128, 135, 152, 159, 167, 180, 188, 191, 200, 207, 212, 215, 216, 239, 240, 263, 276, 284, 288, 303, 311, 320, 335, 344, 348, 359, 368, 375, 380, 383, 384, 392, 404, 423, 431, 447, 456, 464, 479

Offset: 1

Colin Barker, Feb 10 2014

nonn

			15 is in the sequence because x^2 - 10xy + y^2 + 15 = 0 has integer solutions, for example (x, y) = (2, 19).

Cf. A072256 (k = 8), A129445 (k = 15), A080806 (k = 20), A074061 (k = 23), A001079 (k = 24).

Cf. A031363, A084917, A237351, A237599, A237606, A237609.

is(n)=m=bnfisintnorm(bnfinit(x^2-10*x+1),-n);#m>0&&denominator(polcoeff(m[1],1))==1 \\ Ralf Stephan, Feb 11 2014

A236330 Positive integers n such that x^2 - 14xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

32, 48, 128, 176, 192, 288, 368, 416, 432, 512, 624, 704, 752, 768, 800, 944, 1056, 1136, 1152, 1184, 1200, 1328, 1472, 1568, 1584, 1664, 1712, 1728, 1776, 1952, 2048, 2096, 2208, 2288, 2336, 2352, 2496, 2592, 2672, 2816, 2864, 2928, 3008, 3056, 3072, 3104

Offset: 1

Colin Barker, Feb 16 2014

nonn

			48 is in the sequence because x^2 - 14xy + y^2 + 48 = 0 has integer solutions, for example (x, y) = (2, 26).

Cf. A001835 (n = 32), A001075 (n = 48), A237250 (n = 176), A003500 (n = 192), A082841 (n = 288), A151961 (n = 432), A077238 (n = 624).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236331.

A236331 Positive integers n such that x^2 - 18xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

64, 256, 320, 576, 704, 1024, 1216, 1280, 1600, 1856, 1984, 2304, 2624, 2816, 2880, 3136, 3520, 3776, 3904, 4096, 4544, 4864, 5056, 5120, 5184, 5696, 6080, 6336, 6400, 6464, 6976, 7424, 7744, 7936, 8000, 8384, 8896, 9216, 9280, 9536, 9664, 9920, 10496, 10816

Offset: 1

Colin Barker, Feb 16 2014

nonn

			64 is in the sequence because x^2 - 18xy + y^2 + 64 = 0 has integer solutions, for example (x, y) = (1, 13).

Cf. A001519 (n = 64), A052995 (n = 256), A055819 (n = 256), A005248 (n = 320), A237132 (n = 704), A237133 (n = 1216).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330.

A238240 Positive integers n such that x^2 - 20xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

18, 35, 50, 63, 72, 74, 83, 90, 95, 98, 99, 107, 140, 162, 171, 200, 215, 227, 252, 266, 275, 288, 296, 315, 332, 347, 359, 360, 362, 371, 380, 387, 392, 395, 396, 407, 428, 450, 491, 495, 530, 539, 560, 567, 602, 623, 626, 635, 648, 666, 684, 695, 711, 722, 743, 747, 755, 770, 791, 794, 800, 810

Offset: 1

Colin Barker, Feb 20 2014

nonn

Positive integers n such that x^2 - 99 y^2 + n = 0 has integer solutions. - Robert Israel, Oct 22 2024

			63 is in the sequence because x^2 - 20xy + y^2 + 63 = 0 has integer solutions, for example (x, y) = (1, 16).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A075839 (n = 18), A221763 (n = 63), A198947 (n = 90), A001085 (n = 99).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330, A236331.

filter:= t -> [isolve(99*y^2 - z^2 = t)] <> []:
select(filter, [$1..1000]); # Robert Israel, Oct 22 2024

Corrected by Robert Israel, Oct 22 2024

cp's OEIS Frontend

A141123 Primes of the form -x^2+2*x*y+2*y^2 (as well as of the form 3*x^2+6*x*y+2*y^2).

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A142463 a(n) = 2*n^2 + 2*n - 1.

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A237606 Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.

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A237351 Positive integers k such that x^2 - 5xy + y^2 + k = 0 has integer solutions.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

A237599 Positive integers k such that x^2 - 6xy + y^2 + k = 0 has integer solutions.

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Comments

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A237609 Positive integers k such that x^2 - 9xy + y^2 + k = 0 has integer solutions.

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Examples

Crossrefs

Programs

PARI

A237610 Positive integers k such that x^2 - 10xy + y^2 + k = 0 has integer solutions.

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Programs

PARI

A236330 Positive integers n such that x^2 - 14xy + y^2 + n = 0 has integer solutions.

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A236331 Positive integers n such that x^2 - 18xy + y^2 + n = 0 has integer solutions.

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A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).

A142463 a(n) = 2n^2 + 2n - 1.