A031363 -id:A031363 - cp's OEIS frontend

A243188 Nonnegative numbers of the form x^2 + 6xy - 6*y^2.

0, 1, 4, 9, 10, 16, 21, 25, 34, 36, 40, 49, 61, 64, 66, 81, 84, 85, 90, 100, 106, 109, 121, 129, 136, 144, 154, 160, 165, 169, 181, 189, 196, 201, 210, 225, 226, 229, 241, 244, 250, 256, 264, 265, 274, 289, 301, 306, 309, 324, 336, 340, 346, 349, 354, 360

Offset: 1

N. J. A. Sloane, Jun 05 2014

nonn

Discriminant 60.
Also numbers representable as x^2 + 8*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018
Also numbers of the form x^2 - 15*y^2. - Jianing Song, Jul 31 2018

Gheorghe Coserea, Table of n, a(n) for n = 1..100001
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A031363.

Primes: A107152.

Reap[For[n = 0, n <= 200, n++, If[Reduce[ 1*x^2 + 6*x*y - 6*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

More terms from Colin Barker, Apr 07 2015

A022344 Allan Wechsler's "J determinant" sequence.

Original entry on oeis.org

1, 5, 4, 9, 16, 11, 19, 11, 20, 31, 19, 31, 45, 29, 44, 25, 41, 59, 36, 55, 29, 49, 71, 41, 64, 89, 55, 81, 44, 71, 100, 59, 89, 121, 76, 109, 61, 95, 131, 79, 116, 61, 99, 139, 80, 121, 164, 101, 145, 79, 124, 171, 101, 149, 76, 125, 176, 99, 151, 205

Offset: 0

N. J. A. Sloane

nonn

From Kenneth J Ramsey, Jan 06 2007: (Start)
"a(n) = the characteristic value of the row T(n,i) of the Wythoff array A035513 which is the absolute value of T(n,i)^2 - T(n,i-1)*T(n,i+1). Only the number 5 or prime factors ending in 1 or 9 form the squarefree portion of a(n). All other factors of a(n) appear only as squares.
"Moreover, the squarefree portion (less the factor 5) squared is the characteristic value of the Fibonacci sequence whose bijection relates to c term of the Horadam "Fibonacci Number Triples" Amer. Math. Monthly 68(1961) 751-753. That paper showed that if F(0), F(1), F(2), F(3) are 4 sequential numbers in a row of the Wythoff array, then P = (2F(1)*F(2),F(0)*F(1),2F(1)*F(2) + F(0)^2) is a Pythagorean triple (a,b,c) i.e. a^2 + b^2 = c^2.
"If i varies and c(n,2i-1) = F(n,i)^2 + 2F(n,i+1)*F(n,i+2) and C(n,2i) is set to equal C(n,2i+1)-C(n,2i-1) then, the sequence F(x,i) = C(n,i)/G, where G is the greatest common divisor of the adjacent terms C(n,i), is a Fibonacci sequence having the characteristic value which is the square of the squarefree portion of a(n) except without the factor of 5.
"For example the Lucas sequence or the second row of the Wythoff array has the characteristic value of A(2) = 5 and the C(n,i) terms are each 5 times the sequential terms 34,89,233,... which is a bijection of the terms in the 1st row of the Wythoff array which row has the characteristic value of 1. This is so even though adjacent terms of the Lucas sequence are coprime." (End)
Conjecture: Every pair of Fibonacci sequences, F1 and F2, appear in rows n and m of Wythoff's Array, respectively and have respective characteristics a(n) and a(m). Also, there is a third Fibonacci sequence F3, defined by F3(i) = F1(i) * F2(j+1) - F1(i+1)*F2(j) where j is held constant. The sequence F3 appears in row p of Wythoff's array and has the characteristic a(p) = a(n)*a(m). - Kenneth J Ramsey, Feb 11 2007
A product p of rows n and m of the Wythoff Array, such that a(p) = a(n)*a(m) as described in the conjecture above, is defined by A357097(n, m). - Peter Munn, Aug 15 2025
a(n) = |T(n,i)^2 - T(n,i-2)*T(n,i+2)| for all i > 2, where T = Wythoff array.  Indeed, if k > 0, then |T(n,i)^2 - T(n,j-k)*T(n,j+k)| = (F(k)^2)*a(n) for j > k.  That is, if m is in this sequence, then 4*m, 9*m, 25*m, 64*m, ... are also in this sequence.  - Clark Kimberling, Jul 15 2016

Allan C. Wechsler, posting to math-fun mailing list Dec 04 1996.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
A. F. Horadam, Fibonacci Number Triples, The American Mathematical Monthly, Vol. 68, No. 8 (Oct., 1961), pp. 751-753.

Cf. A035513, A127561, A275068, A357097.

Range of values: A031363.

[Floor((n+1)*((1+Sqrt(5))/2))^2-n*Floor((n+1)*(1+Sqrt(5))/2)-n^2: n in [0..60]]; // Vincenzo Librandi, Jul 01 2016

Digits := 50: t := evalf((1+sqrt(5))/2): f := n->floor( n*t)^2-(n-1)*floor(n*t)-(n-1)^2:

Table[#^2 - n # - n^2 &[Floor[(n + 1) GoldenRatio]], {n, 0, 51}] (* Michael De Vlieger, Jun 30 2016 *)

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

a(A357097(x,y)) = a(x) * a(y). - Peter Munn, Aug 15 2025

A035282 Expansion of zeta function of icosian ring (nonzero terms).

Original entry on oeis.org

1, 5, 6, 10, 24, 21, 40, 30, 31, 60, 64, 50, 84, 120, 60, 50, 144, 120, 124, 85, 144, 200, 160, 126, 91, 180, 240, 240, 155, 204, 220, 300, 410, 320, 156, 264, 280, 210, 360, 300, 304, 384, 420, 170, 400, 504, 360, 300, 364, 384, 250, 400, 504, 960, 424, 720, 310

Offset: 1

N. J. A. Sloane

Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187; then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

Nonzero terms of A078473. - Michel Marcus, Mar 03 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. (1999), 51 1258-1276.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.

Cf. A031363, A035187, A078472, A078473.

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &] (* Amiram Eldar, May 13 2022 *)

A078473 Expansion of zeta function of icosian ring.

Original entry on oeis.org

1, 0, 0, 5, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 21, 0, 0, 40, 30, 0, 0, 0, 0, 31, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 50, 0, 0, 0, 0, 84, 0, 0, 120, 60, 0, 0, 0, 50, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 85, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 200, 0, 0, 160, 126, 91, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Benoit Cloitre, Dec 31 2002

Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187. Then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276; arXiv preprint, arXiv:math/9904028 [math.MG], 1999.

Cf. A035187, A035282 (nonzero terms of the sequence), A031363 (n for which a(n) is not zero), A078471.

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 13 2022 *)

{a(n)=local(A); if(n<1, 0, A=direuler(p=2,n,1/(1-X)/(1-kronecker(5,p)*X)); sumdiv(n,d,A[d]*d*A[n/d]))} /* Michael Somos, Jun 06 2005 */

pf(p, r) = {if (p==5, (5^(r+1) -1)/4, if (((p % 5) == 2) || ((p % 5) == 3), if (!(r % 2), (p^(r+2) - 1)/(p^2-1), 0), if (((p % 5) == 1) || ((p % 5) == 4), sum(k=0, r, (k+1)*(r-k+1)*p^k))););}
a(n) = {my(f = factor(n)); prod(i=1, #f~, pf(f[i, 1], f[i, 2]));} \\ Michel Marcus, Mar 03 2014

Multiplicative with a(p^e) = (5^(e + 1) - 1)/4 if p = 5, (p^(e + 2) - 1)/(p^2 - 1) or 0 if p == 2 or 3 (mod 5) and e is even or odd, respectively, and Sum_{k=0..e} (k + 1)*(e - k + 1)*p^k if p == 1 or 4 (mod 5). - Amiram Eldar, May 13 2022

A232681 Numbers n such that the equation a^2 + 5nb^2 = 5c^2 + nd^2 has no solutions in positive integers for a, b, c, d.

Original entry on oeis.org

2, 3, 6, 7, 8, 10, 12, 13, 14, 15, 17, 18, 21, 22, 23, 24, 26, 27, 28, 30, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 60, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 97, 98

Offset: 1

V. Raman, Nov 27 2013

nonn

With n = 2, the equation a^2 + 10*b^2 = 2*d^2 + 5*c^2 has no solutions in positive integers for a, b, d, c as the following proof shows: Let's assume that gcd(a, b, d, c) = 1, otherwise if gcd(a, b, d, c) = g, then a/g, b/g, d/g, c/g would be a smaller set of solutions to the equation. Considering modulo 5 arithmetic, we have a^2 - 2*d^2 == 0 (mod 5). Since a square is always congruent to 0 (mod 5), 1 (mod 5) or 4 (mod 5), this is possible if and only if a == 0 (mod 5) and d == 0 (mod 5). Now let a = 5*p, d = 5*q, so a^2 = 25*p^2, d^2 = 25*q^2. Substituting this into the equation a^2 + 10*b^2 = 2*d^2 + 5*c^2 gives 25*p^2 + 10*b^2 = 50*q^2 + 5*c^2, i.e. 5*p^2 + 2*b^2 = 10*q^2 + c^2. Taking modulo 5 arithmetic with this equation again gives 2*b^2 - c^2 == 0 (mod 5). By using the same argument as above, this is possible if and only if b == 0 (mod 5) and c == 0 (mod 5). We already showed that a == 0 (mod 5) and d == 0 (mod 5), so gcd(a, b, d, c) should be a multiple of 5. This contradicts our assumption that gcd(a, b, d, c) = 1 and a/5, b/5, d/5, c/5 are a smaller set of solutions to the above mentioned equation. By using the proof of infinite descent, this implies that the only possible set of solutions to (a, b, d, c) is (0, 0, 0, 0).
We can similarly prove for the other values of n by taking modulo 5 arithmetic if the only solution to a^2 - n*d^2 == 0 (mod 5) is a == 0 (mod 5) and d == 0 (mod 5). This happens if n == 2, 3 (mod 5).
On the other hand, if we take modulo n arithmetic and if a^2 - 5*d^2 == 0 (mod n) has the only solution a == 0 (mod n) and d == 0 (mod n), then n is a member of this sequence. If r is a prime factor of n and if r^2 does not divide n and the equation a^2 - 5*d^2 == 0 (mod r) has the only solution a == 0 (mod r) and d == 0 (mod r), we can also take modulo r arithmetic to prove that n is a member of this sequence.
If n = 5*k is a multiple of 5 and not a multiple of 25, taking modulo 5 arithmetic yields 'a' to be a multiple of 5. Putting a = 5*p, and dividing the equation by 5 gives 5*(p^2+k*b^2) = (c^2+k*d^2). This equation will have no solution in positive integers p, b, c, d if and only if there is no number that can be written by the form x^2+k*y^2 that is 5 times another number that can be written by the same form x^2+k*y^2.
If n is a multiple of 25, then n = 25*m is a member of this sequence if and only if m is a member of this sequence.
This appears to be the complement of A031363. If so, the definition could be simplified. - Franklin T. Adams-Watters, Apr 02 2016
This is the complement of A031363. Proof: From the equation in the name follows a^2 - 5c^2 = n(d^2 - 5b^2). This equation has positive integer solutions if n is of the form x^2 - 5y^2, because A031363 is closed under multiplication.  If there is no positive integer solution for the equation, it is because n is not a member of A031363. Thus n belongs to the present sequence, which was to be proved. This sequence contains no squares, but all odd powers of a term belong to the sequence. - Klaus Purath, Jul 31 2023

			n = 2 is a member of this sequence because there is no positive integer m which can be simultaneously written as both x^2+10*y^2 and 5*x^2+2*y^2. The former requires the sum of {2, 5, 7, 13, 23, 37} mod 40 prime factors of m to be even, while the latter requires the sum of {2, 5, 7, 13, 23, 37} mod 40 prime factors of m to be odd.
n = 3 is a member of this sequence because there is no positive integer m which can be simultaneously written as both x^2+15*y^2 and 5*x^2+3*y^2. The former requires the sum of {2, 3, 5, 8} mod 15 prime factors of m to be even, while the latter requires the sum of {2, 3, 5, 8} mod 15 prime factors of m to be odd.

V. Raman, Proof for individual terms

Cf. A031363, A232531, A232532, A232682.

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.

A242663 Nonnegative integers of the form x^2 + 4xy - 4*y^2.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 17, 25, 28, 32, 36, 41, 49, 56, 64, 68, 72, 73, 81, 89, 92, 97, 100, 112, 113, 121, 124, 128, 136, 137, 144, 153, 161, 164, 169, 184, 188, 193, 196, 200, 217, 224, 225, 233, 241, 248, 252, 256, 257, 272, 281, 284, 288, 289, 292

Offset: 1

N. J. A. Sloane, May 31 2014

nonn

Discriminant 32.
Also numbers representable as x^2 + 6*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018
Also numbers of the form x^2 - 8*y^2. - Jianing Song, Jul 31 2018

Gheorghe Coserea, Table of n, a(n) for n = 1..100001
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A031363.

Primes in this sequence = A007519.

Reap[For[n = 0, n <= 300, n++, If[Reduce[ x^2 + 4*x*y - 4*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

seq(M, k=6) = {
setintersect([1..M], setbinop((x, y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
concat(0, seq(292)) \\ Gheorghe Coserea, Jul 31 2018

A243172 Nonnegative integers of the form x^2 + 3xy - 3*y^2 (discriminant 21).

Original entry on oeis.org

0, 1, 4, 7, 9, 15, 16, 25, 28, 36, 37, 43, 49, 51, 60, 63, 64, 67, 79, 81, 85, 100, 105, 109, 112, 121, 123, 127, 135, 141, 144, 148, 151, 163, 169, 172, 175, 177, 193, 196, 204, 205, 211, 225, 235, 240, 249, 252, 256, 259, 267, 268, 277, 289, 295, 301, 303, 316, 324, 331, 333, 337, 340, 343, 357, 361, 373, 375, 379, 387, 393, 400, 415, 420, 421, 436, 441, 445, 448, 457, 459, 463, 469, 484, 487, 492, 499

Offset: 1

N. J. A. Sloane, Jun 01 2014

nonn

Also numbers representable as x^2 + 5*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018
Also numbers of the form x^2 - x*y - 5*y^2 with 0 <= x <= y (or x^2 + x*y - 5*y^2 with x, y nonnegative). - Jianing Song, Jul 31 2018
Also nonnegative numbers of the form 7*x^2 - 3*y^2. - Jon E. Schoenfield, Jun 03 2022

Gheorghe Coserea, Table of n, a(n) for n = 1..100001
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A031363.

Primes: A141159.

A={}; For[ n=0, n <= 300, n++,
    If[ Length[ Reduce[x^2 + 3 x y - 3 y^2 - n == 0, {x,y}, Integers]]>0, AppendTo[A,n]]]; A

\\ From Bill Allombert, Jun 04 2014. Since 21 is a fundamental discriminant, and the polynomial is unitary, the following code works:
B=bnfinit(x^2+3*x-3); select(n->#bnfisintnorm(B,n),[1..500])

A243174 Nonnegative integers of the form x^2 + 5xy - 5*y^2 (discriminant 45).

Original entry on oeis.org

0, 1, 4, 9, 16, 19, 25, 31, 36, 45, 49, 55, 61, 64, 76, 79, 81, 99, 100, 109, 121, 124, 139, 144, 145, 151, 169, 171, 180, 181, 196, 199, 205, 211, 220, 225, 229, 241, 244, 256, 261, 271, 279, 289, 295, 304, 316, 319, 324, 331, 349, 355, 361, 369, 379, 396, 400, 405, 409, 421, 436, 439, 441, 445, 451, 475, 484, 495, 496, 499, 505, 529, 531, 541, 549

Offset: 1

N. J. A. Sloane, Jun 01 2014

nonn

Also numbers representable as x^2 + 7*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018
Also numbers of the form x^2 - x*y - 11*y^2 with 0 <= x <= y (or x^2 + x*y - 11*y^2 with x, y nonnegative). - Jianing Song, Jul 31 2018
Also nonnegative integers of the form 9x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022

Gheorghe Coserea, Table of n, a(n) for n = 1..100001
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A028957, A031363.

Primes: A033212.

cp's OEIS Frontend

A243188 Nonnegative numbers of the form x^2 + 6*x*y - 6*y^2.

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A022344 Allan Wechsler's "J determinant" sequence.

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Magma

Maple

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A035282 Expansion of zeta function of icosian ring (nonzero terms).

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A078473 Expansion of zeta function of icosian ring.

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PARI

PARI

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A232681 Numbers n such that the equation a^2 + 5*n*b^2 = 5*c^2 + n*d^2 has no solutions in positive integers for a, b, c, d.

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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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A242663 Nonnegative integers of the form x^2 + 4*x*y - 4*y^2.

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Mathematica

PARI

A243172 Nonnegative integers of the form x^2 + 3*x*y - 3*y^2 (discriminant 21).

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A243188 Nonnegative numbers of the form x^2 + 6xy - 6*y^2.

A232681 Numbers n such that the equation a^2 + 5nb^2 = 5c^2 + nd^2 has no solutions in positive integers for a, b, c, d.

A242663 Nonnegative integers of the form x^2 + 4xy - 4*y^2.

A243172 Nonnegative integers of the form x^2 + 3xy - 3*y^2 (discriminant 21).

A243174 Nonnegative integers of the form x^2 + 5xy - 5*y^2 (discriminant 45).