A047201 -id:A047201 - cp's OEIS frontend

A330133 a(n) = (1/16)(5 + (-1)^(1+n) - 4cos(nPi/2) + 10n^2).

0, 1, 3, 6, 10, 16, 23, 31, 40, 51, 63, 76, 90, 106, 123, 141, 160, 181, 203, 226, 250, 276, 303, 331, 360, 391, 423, 456, 490, 526, 563, 601, 640, 681, 723, 766, 810, 856, 903, 951, 1000, 1051, 1103, 1156, 1210, 1266, 1323, 1381, 1440, 1501, 1563, 1626, 1690, 1756

Offset: 0

Stefano Spezia, Dec 02 2019

For n > 0, partial sums of A047201.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A005891, A033583 (10*n^2), A047201.

I:=[0, 1, 3, 6, 10, 16]; [n le 6 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..54]];

gf:=(1/16)*(-exp(-x) + 5*exp(x)*(1 + 2*x + 2*x^2) - 4*cos(x)); ser := series(gf, x, 54):
seq(factorial(n)*coeff(ser, x, n), n = 0 .. 53)

Table[(1/16)*(5+(-1)^(1+n)-4*Cos[n*Pi/2]+10*n^2),{n,0,53}]
LinearRecurrence[{2,-1,0,1,-2,1},{0,1,3,6,10,16},60] (* Harvey P. Dale, Jul 21 2021 *)

concat([0], Vec(-x*(1 + x + x^2 + x^3 + x^4)/((-1 + x)^3*(1 + x)*(1 + x^2))+O(x^54)))

O.g.f.: -x*(1 + x + x^2 + x^3 + x^4)/((-1 + x)^3*(1 + x)*(1 + x^2)).
E.g.f.: (1/16)*(-exp(-x) + 5*exp(x)*(1 + 2*x + x^2) - 4*cos(x)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) -2*a(n-5) + a(n-6) for n > 5.
a(2*n-1) = A005891(n-1) for n > 0.
a(4*n) = 10*n^2. - Bernard Schott, Dec 06 2019

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, #] &]

cp's OEIS Frontend

A330133 a(n) = (1/16)(5 + (-1)^(1+n) - 4cos(nPi/2) + 10n^2).

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

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cp's OEIS Frontend

A330133 a(n) = (1/16)*(5 + (-1)^(1+n) - 4*cos(n*Pi/2) + 10*n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

A330133 a(n) = (1/16)(5 + (-1)^(1+n) - 4cos(nPi/2) + 10n^2).

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