XU Pingya - cp's OEIS frontend

A356717 a(n) is the integer w such that (c(n)^2, -d(n)^2, w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+3) + (-1)^n * F(n-2) and F(n) is the n-th Fibonacci number (A000045).

1, 29, 59, 241, 445, 1691, 3089, 11629, 21211, 79745, 145421, 546619, 996769, 3746621, 6831995, 25679761, 46827229, 176011739, 320958641, 1206402445, 2199883291, 8268805409, 15078224429, 56675235451, 103347687745, 388457842781, 708355589819, 2662529664049

Offset: 1

XU Pingya, Aug 24 2022

			For n=3, 2 * ((F(5) - F(0))^2)^3 + 2 * (-(F(6) - F(1))^2)^3 + 59^3 = 2 * 25^3 - 2 * 49^3 + 59^3 = 1331, a(3) = 59.

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-1,1).

Cf. A000045, A081016, A081018, A089270, A228208, A237132.

Cf. also A337929, A354337, A356716.

Table[(1331-2*((Fibonacci[n+2]+(-1)^n*Fibonacci[n-3]))^6+2*(Fibonacci[n+3]+(-1)^n*Fibonacci[n-2])^6)^(1/3), {n,28}]

a(n) = (1331 - 2 * A237132(n)^6 + 2 * A228208(n+1)^6)^(1/3).
a(n) = ((1-(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n-1} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)) + ((1+(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)).
a(n) = ((1-(-1)^n)/2) * (-5 + 14 * A081018(n-1) + 6 * A081016(n-1)) + ((1+(-1)^n)/2) * (-5 + 14 * A081018(n) + 6 * A081016(n-1)).
From Stefano Spezia, Aug 25 2022: (Start)
G.f.: x*(1 + 28*x + 23*x^2 - 14*x^3 - 5*x^4)/((1 - x)*(1 - 3*x + x^2)*(1 + 3*x + x^2)).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)

A356716 a(n) is the integer w such that (c(n)^2, -d(n)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+1) + (-1)^n * F(n-4) and F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

5, 19, 31, 101, 179, 655, 1189, 4451, 8111, 30469, 55555, 208799, 380741, 1431091, 2609599, 9808805, 17886419, 67230511, 122595301, 460804739, 840280655, 3158402629, 5759369251, 21648013631, 39475304069, 148377692755, 270567759199, 1016995835621, 1854499010291

Offset: 1

XU Pingya, Aug 24 2022

Conjecture:
(i) For all k > 2, 2*x^3 + 2*y^3 + z^3 = A089270(k)^3 have primitive solutions form (c(n)^2, -d(n)^2, -w(n)) with d(n) = 3*d(n-2) - d(n-4), c(n) = d(n+2) - d(n) and w(n) = 8*w(n-2) - 8*w(n-4) + w(n-6).
(ii) This sequence is a subsequence of A089270.
From XU Pingya, Jun 07 2024: (Start)
Several positive examples of conjecture:
When A089270(4,5,6,7) = {19,29,31,41}, d(n) can be taken as:
(1/2) * (F(n+3) + (-1)^n * F(n-6));
((1-(-1)^n)/2) * (F(n+3) + F(n-4)) + ((1+(-1)^n)/2) * (F(n+3) - F(n-4));
((1-(-1)^n)/2) * (2*F(n-1) + 3*F(n-3)) + ((1+(-1)^n)/2) * (3*F(n-2) + 2*F(n-4));
and
((1-(-1)^n)/2) * (2*F(n+1) + F(n-5)) + ((1+(-1)^n)/2) * (F(n+2) + 2*F(n-4)).
When A089270(17) = 121, d(n) can be taken as d(1,2,3,4) = {-3,0,7,11}. (End)
From XU Pingya, Jul 17 2024: (Start)
Furthermore, we observe that if (x, y) (y < x/2) is the solution of the Diophantine equation x^2 + x * y - y^2 = A089270(k). Let
d(2*n-1) = x * F(2*n-2) - y * F(2*n-3), c(2*n-1) = d(2*n+1) - d(2*n-1);
d(2*n) = x * F(2*n-2) + y * F(2*n-1), c(2*n) = d(2*n+2) - d(2*n).
Then such c(n) and d(n) satisfy the conjecture. (End)

			For n=3, 2 * ((F(5) - F(0))^2)^3 + 2 * (-(F(4) - F(-1))^2)^3 + (-31)^3 = 2 * 25^3 - 2 * 4^3 - 31^3 = 1331, a(3) = 31.

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-1,1).

Cf. A000045, A081016, A089270, A206351, A228208, A237132.

Cf. also A337928, A354336, A356717.

Table[(-1331+2*((Fibonacci[n+2]+(-1)^n*Fibonacci[n-3]))^6-2*(Fibonacci[n+1]+(-1)^n*Fibonacci[n-4])^6)^(1/3), {n,28}]

a(n) = (-1331 + 2 * A237132(n)^6 - 2 * A228208(n-1)^6)^(1/3).
a(n) = ((1-(-1)^n)/2) * (-1 + 6 * Sum_{k=0..n-1} Fibonacci(4*k-1) + 14 * Sum_{k=0..n-2} Fibonacci(4*k+1)) + ((1+(-1)^n)/2) * (-1 + 6 * Sum_{k=0..n-1} Fibonacci(4*k-1) + 14 * Sum_{k=0..n-1} Fibonacci(4*k+1)).
a(n) = ((1-(-1)^n)/2) * (-1 + 6*A206351(n) + 14*A081016(n-2)) + ((1+(-1)^n)/2) * (-1 + 6*A206351(n) + 14*A081016(n-1)).
From Stefano Spezia, Aug 25 2022: (Start)
G.f.: x*(5 + 14*x - 23*x^2 - 28*x^3 - x^4)/((1 - x)*(1 - 3*x + x^2)*(1 + 3*x + x^2)).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
From XU Pingya, Jul 17 2024: (Start)
a(2*n-1) = (F(2*n) + F(2*n-2) + F(2*n-5))^2 + (F(2*n) + F(2*n-2) + F(2*n-5)) * (F(2*n-2) + F(2*n-4) + F(2*n-7)) - (F(2*n-2) + F(2*n-4) + F(2*n-7))^2;
a(2*n) = (F(2*n+2) + F(2*n-3))^2 + (F(2*n+2) + F(2*n-3)) * (F(2*n) + F(2*n-5)) - (F(2*n) + F(2*n-5))^2. (End)

A354337 a(n) is the integer w such that (L(2n)^2, -L(2n + 1)^2, w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

Original entry on oeis.org

19, 149, 1039, 7139, 48949, 335519, 2299699, 15762389, 108037039, 740496899, 5075441269, 34787591999, 238437702739, 1634276327189, 11201496587599, 76776199786019, 526231901914549, 3606847113615839, 24721697893396339, 169445038140158549, 1161393569087713519

Offset: 1

XU Pingya, Jun 20 2022

Subsequence of A017377.

			2*(L(4)^2)^3 + 2*(-L(5)^2)^3 + (149)^3 = 2*(49)^3 + 2*(-121)^3 + (149)^3 = 125, a(2) = 149.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000032, A002878, A005248, A017377, A081017, A089508, A092521, A288913.

Cf. A337929, A354336.

LinearRecurrence[{7,-1},{19,149},21]-1 + LucasL[2*Range[21]-3]^2

a(n) = (125 - 2*A005248(n)^6 + 2*A002878(n)^6)^(1/3).
a(n) = Lucas(4*n+2) + Lucas(4n-1) - 3 = 2*A056914(n)-3 = 15*A092521(n) + A288913(n-1).
a(n) = 2*A081017(n) - 1.
a(n) = 10*A089508(n) + 9.
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: x*(19 - 3*x - x^2)/((1 - x)*(1 - 7*x + x^2)). - Stefano Spezia, Jun 22 2022

A354336 a(n) is the integer w such that (L(2n)^2, -L(2n-1)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

Original entry on oeis.org

1, 11, 61, 401, 2731, 18701, 128161, 878411, 6020701, 41266481, 282844651, 1938646061, 13287677761, 91075098251, 624238009981, 4278590971601, 29325898791211, 201002700566861, 1377693005176801, 9442848335670731, 64722245344518301, 443612869075957361

Offset: 0

XU Pingya, Jun 20 2022

Subsequence of A017281.

			2*(L(4)^2)^3 + 2*(-L(3)^2)^3 + (-61)^3 = 2*(49)^3 + 2*(-1)^3 + (-61)^3 = 125, a(2) = 61.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000032, A002878, A005248, A017281, A056914, A081015, A092521.

Cf. A337928, A354337.

LucasL[4*Range[22]-3] + 1 - LucasL[2*Range[22]-3]^2

a(n) = (-125 + 2*A005248(n)^6 - 2*A002878(n-1)^6)^(1/3).
a(n) = Lucas(4*n+1) - Lucas(4*n-2) + 3 = A056914(n) - 15*A092521(n-1), for n > 1.
a(n) = Lucas(4*n+1) + 1 - Lucas(2*n-1)^2.
a(n) = 2*A081015(n-1) + 1.
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (1 + 3*x - 19*x^2)/((1 - x)*(1 - 7*x + x^2)). - Stefano Spezia, Jun 22 2022
a(n) = (F(2*n+1) + F(2*n-1))^2 + (F(2*n+1) + F(2*n-1)) * (F(2*n-1) + F(2*n-3)) - (F(2*n-1) + F(2*n-3))^2. - XU Pingya, Jul 17 2024

A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

Original entry on oeis.org

2, 16, 128, 1024, 1458, 8192, 11664, 31250, 65536, 93312, 235298, 524288, 746496, 1062882, 2000000, 3543122, 3906250, 5971968, 9653618, 15059072, 22781250, 28697814, 33554432, 47775744, 48275138, 68024448, 80707214, 94091762, 128000000, 171532242, 226759808

Offset: 1

XU Pingya, Nov 16 2020

nonn

The data are derived from the following formula:
(a^2 - a*t - t^2)^3 + (a^2 + a*t - t^2)^3 + 2*(t^2)^3 = 2*a^6
(a^3 - 3*t^3)^3 + (a^3 + 3*t^3) + 2*(-3*a*t^2)^3 = 2*a^9;
(9*a^3 - t^3)^3 + (9*a^3 + t^3)^3 + 2*(-3*a*t^2)^3 = 1458*a^9;
(6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;
(6*a^3*t - 9*t^4)^3 + (9*t^4)^3 + 2*(2*a^4 - 9*a*t^3)^3 = 16*a^12 = 2*2^3*a^12;
(18*a^3*t - 8*t^4)^3 + (8*t^4)^3 + 2*(9*a^4 - 12*a*t^3)^3 = 1458*a^12 = 2*9^3*a^12;
(18*a^3*t - t^4)^3 + (t^4)^3 + 2*(18*a^4 - 3*a*t^3)^3 = 11664*a^12 = 2*18^3*a^12.

			16 is a term, because when t is an integer, (6*(2*t + 1) - 9*(2*t + 1)^4, 9*(2*t + 1)^4, 2 - 9*(2*t + 1)^3) is a nontrivial primitive parametric solution of x^3 + y^3 + 2*z^3 = 16.

R. K. Guy, Unsolved Problems in Number Theory, D5.

Kenji Koyama, On searching for solutions of the Diophantine equation x^3 + y^3 + 2z^3 = n, Math. Comp, 69 (2000), 1735-1742.
J. C. P. Miller & M. F. C. Woollett, Solutions of the Diophantine equation x^3 + y^3 + z^3 = k, J. London Math. Soc. 30(1955), 101-110.
Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338932.

t1 = 2*Range[23]^6;
t2 = 2*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*Range[4]^9;
t4 = 2*{1, 5}^12;
t5 = 16*{1, 2, 4}^12;
t6 = 1458*{1, 3}^12;
t7 = 11664*{1, 2, 3}^12;
Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]

Missing terms 1024 and 746496 added by XU Pingya, Mar 14 2022

A338932 Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.

Original entry on oeis.org

1, 2, 128, 729, 1458, 4096, 65536, 93312, 2985984, 3906250, 16777216, 28697814, 33554432, 47775744, 80707214, 244140625, 250000000, 387420489, 1836660096, 2847656250, 4715895382, 5165261696, 12230590464, 13841287201, 17179869184, 21208998746, 24461180928

Offset: 1

XU Pingya, Nov 16 2020

nonn

The data are derived from the following formula:
(a^3 - 6*t^3)^3 + (a^3 + 6*t^3)^3 + (-6*a*t^2)^3 = 2*a^9;
(4*a^3 - 3*t^3)^3 + (4*a^3 + 3*t^3)^3 + (-6*a*t^2)^3 = 128*a^9 = 2*4^3*a^9;
(9*a^3 - 2*t^3)^3 + (9*a^3 + 2*t^3)^3 + (-6*a*t^2)^3 = 1458*a^9 = 2*9^3*a^9;
(36*a^3 - t^3)^3 + (36*a^3 + t^3)^3 + (-6*a*t^2)^3 = 93312*a^9 = 2*36^3*a^9;
((3*a^3)*t - 9*t^4)^3 + (9*t^4)^3 + (a^4 - 9*a*t^3)^3 = a^12;
((9*a^3)*t - t^4)^3 + (t^4)^3 + (9*a^4 - 3*a*t^3)^3 = 729*a^12 = 9^3*a^12.

			128 is a term, because (4 - 3*(2*n - 1)^3, 4 + 3*(2*n - 1)^3, -3*(2*n - 1)^2) is a nontrivial primitive parametric solution of x^3 + y^3 + z^3 = 128.

R. K. Guy, Unsolved Problems in Number Theory, D5.

Kenji Koyama, On searching for solutions of the Diophantine equation x^3 + y^3 + 2z^3 = n, Math. Comp, 69 (2000), 1735-1742.
J. C. P. Miller & M. F. C. Woollett, Solutions of the Diophantine equation x^3 + y^3 + z^3 = k, J. London Math. Soc. 30(1955), 101-110.
Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338933.

t1 = 2*{1, 5, 7, 11, 13}^9;
t2 = 128*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*{1, 3, 5, 7, 9}^9;
t4 = 93312*{1, 2, 3, 4, 5}^9;
t5 = {1, 2, 4, 5, 7}^12;
t6 = 729*{1, 2, 3, 4, 5}^12;
Take[Union[t1, t2, t3, t4, t5, t6], 27]

A338239 Values z of primitive solutions (x, y, z) to the Diophantine equation 2x^3 + 2y^3 + z^3 = 1.

Original entry on oeis.org

-1, 1, -5, 11, -17, 19, 29, -31, -37, -61, 79, -85, 113, -127, -143, 145, -209, 305, 361, -485, 487, 545, 647, 667, 811, -1091, -1151, 1153, -1235, -1429, -1525, 1597, 1699, -1793, -2249, 2251, -2533, 2627, -2677, 2977, -2981, 3089, -3295, 3739, -3887, 3889

Offset: 1

XU Pingya, Oct 18 2020

sign

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

When x = (3*c)*t - (9*a)*t^4, y = (9*a)*t^4, z = c - (9*a)*t^3; a*x^3 + a*y^3 + c*z^3 = c^4. Let a = 2, c = 1, then 1 - 18*n^3 and 1 + 18*n^3 are terms of the sequence. Also, -A337928 and A337929 are subsequences.

			2*25^3 + 2*(-64)^3 + 79^3 = 2*164^3 + 2*(-167)^3 + 79^3 = 1, 79 is a term.

Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.

Cf. A000045, A000290, A000578, A003215, A004825, A004826, A047201, A050791, A130472, A195006, A337928, A337929.

Clear[t]
t = {};
Do[y = ((1 - 2x^3 - z^3)/2)^(1/3) /. (-1)^(1/3) -> -1;
 If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -4000, 4000}, {x, -Round[(Abs[1 + z^3]/6)^(1/2)], Round[(Abs[1 + z^3]/6)^(1/2)]}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 8001}];
Select[v, MemberQ[u, #] &]

A337929 Numbers w such that (F(2n-1)^2, -F(2n)^2, w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

1, 11, 79, 545, 3739, 25631, 175681, 1204139, 8253295, 56568929, 387729211, 2657535551, 18215019649, 124847601995, 855718194319, 5865179758241, 40200540113371, 275538601035359, 1888569667134145, 12944449068903659, 88722573815191471, 608113567637436641

Offset: 1

XU Pingya, Sep 30 2020

			2*(F(3)^2)^3 + 2*(-F(4)^2)^3 + 11^3 = 2*4^3 + 2*(-9)^3 + 11^3 = 1, 11 is a term.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A000578, A003482, A056573, A337928.

Table[(2*Fibonacci[2n]^6 - 2*Fibonacci[2n-1]^6 + 1)^(1/3), {n, 22}]
LinearRecurrence[{8,-8,1},{1,11,79},30] (* Harvey P. Dale, Aug 23 2021 *)

a(n) = (2*F(2*n)^6 - 2*F(2*n-1)^6 + 1)^(1/3).
From Colin Barker, Oct 01 2020: (Start)
G.f.: x*(1 + 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>3.
(End)
a(n) = 2*A003482(n) + 1. - Hugo Pfoertner, Oct 01 2020
a(n) = A033888(n) - A064170(n+2). - Flávio V. Fernandes, Jan 10 2021

A337928 Numbers w such that (F(2n+1)^2, -F(2n)^2, -w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

1, 5, 31, 209, 1429, 9791, 67105, 459941, 3152479, 21607409, 148099381, 1015088255, 6957518401, 47687540549, 326855265439, 2240299317521, 15355239957205, 105246380382911, 721369422723169, 4944339578679269, 33889007628031711, 232278713817542705

Offset: 0

XU Pingya, Sep 30 2020

			2*(F(5)^2)^3 + 2*(-F(4)^2)^3 + (-31)^3 = 2*(25)^3 + 2*(-9)^3 + (-31)^3 = 1, a(2) = 31.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A000578, A056573, A081018, A337929.

Table[(2*Fibonacci[2n+1]^6 - 2*Fibonacci[2n]^6 - 1)^(1/3), {n, 0, 21}]
Table[(Fibonacci[2n+1]*Fibonacci[2n+2]- Fibonacci[2n]^2), {n, 0, 21}] (* Wolfgang Berndt, May 26 2023 *)
LinearRecurrence[{8,-8,1},{1,5,31},30] (* Harvey P. Dale, Dec 17 2023 *)

Vec((1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)) + O(x^20)) \\ Colin Barker, Oct 01 2020

a(n) = (2*F(2*n+1)^6 - 2*F(2*n)^6 - 1)^(1/3).
From Colin Barker, Oct 01 2020: (Start)
G.f.: (1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>2.
(End)
a(n) = 2*A081018(n) + 1. - Hugo Pfoertner, Oct 01 2020
a(n) = A064170(n+2) + A033888(n). - Flávio V. Fernandes, Jan 10 2021
a(n) = F(2*n+1)*F(2*n+2) - F(2*n)^2. - Wolfgang Berndt, May 26 2023
a(2*n-1) = 5 + 6*Sum_{k=1..n-1} F(8*k+1), a(2*n) = 1 + 6*Sum_{k=1..n} F(8*k-3). - XU Pingya, Jun 09 2024

A336451 Primes of form x^3 - (x + 1)^3 + 3z^3 or -x^3 + (x + 1)^3 - 3z^3, with x,z >= 0.

Original entry on oeis.org

2, 5, 7, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 73, 79, 101, 103, 107, 113, 127, 131, 139, 149, 173, 179, 181, 191, 193, 199, 251, 263, 269, 271, 277, 307, 317, 331, 367, 373, 379, 383, 389, 397, 431, 439, 479, 503, 509, 521, 523, 547, 557, 563, 569, 571

Offset: 1

XU Pingya, Aug 31 2020

nonn

For z <= 10^6, no other prime have this form in the first 105 primes.

			0^3 - 1^3 + 3*2^3 = 23, 23 is a term.
-3^3 + 4^3 - 3*0^3 = -4^3 + 5^3 - 3*2^3 = -52^3 + 53^3 - 3*14^3 = 37, 37 is a term.

Cf. A007490, A221794.

p1 = Select[Prime[Range[105]], IntegerQ[(# - 1)/3] &];
p2 = Select[Prime[Range[105]], IntegerQ[(# + 1)/3] &];
n1 = Length@p1; n2 = Length@p2;
r1 = (p1 - 1)/3; r2 = (p2 + 1)/3;
t = {};
Do[x = (z^3 + r1[[n]] + 1/4)^(1/2) - 1/2;
 If[IntegerQ[x], AppendTo[t, -x^3 + (x + 1)^3 - 3z^3]], {n, 1,
  n1}, {z, 0, 270}]
Do[x = (z^3 - r2[[n]] + 1/4)^(1/2) - 1/2;
 If[IntegerQ[x], AppendTo[t, x^3 - (x + 1)^3 + 3z^3]], {n, 1,
  n2}, {z, 0, 170}]
Union@t

cp's OEIS Frontend

User: XU Pingya

A356717 a(n) is the integer w such that (c(n)^2, -d(n)^2, w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+3) + (-1)^n * F(n-2) and F(n) is the n-th Fibonacci number (A000045).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A356716 a(n) is the integer w such that (c(n)^2, -d(n)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+1) + (-1)^n * F(n-4) and F(n) is the n-th Fibonacci number (A000045).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A354337 a(n) is the integer w such that (L(2*n)^2, -L(2*n + 1)^2, w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A354336 a(n) is the integer w such that (L(2*n)^2, -L(2*n-1)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A338932 Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A338239 Values z of primitive solutions (x, y, z) to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A337929 Numbers w such that (F(2*n-1)^2, -F(2*n)^2, w) are primitive solutions of the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

Views

A356717 a(n) is the integer w such that (c(n)^2, -d(n)^2, w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+3) + (-1)^n * F(n-2) and F(n) is the n-th Fibonacci number (A000045).

A356716 a(n) is the integer w such that (c(n)^2, -d(n)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+1) + (-1)^n * F(n-4) and F(n) is the n-th Fibonacci number (A000045).

A354337 a(n) is the integer w such that (L(2n)^2, -L(2n + 1)^2, w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

A354336 a(n) is the integer w such that (L(2n)^2, -L(2n-1)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

A338239 Values z of primitive solutions (x, y, z) to the Diophantine equation 2x^3 + 2y^3 + z^3 = 1.

A337929 Numbers w such that (F(2n-1)^2, -F(2n)^2, w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

A337928 Numbers w such that (F(2n+1)^2, -F(2n)^2, -w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

A336451 Primes of form x^3 - (x + 1)^3 + 3z^3 or -x^3 + (x + 1)^3 - 3z^3, with x,z >= 0.