A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

0, 1, 1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489

Offset: 0

Author

Jean-François Alcover and Paul Curtz, Jul 19 2017

Keywords

• sign
• easy

Comments

This is the inverse binomial transform of A157241.

Successive differences of A157241 begin:

0, 1, 3, 3, -5, -21, -21, 43, 171, 171, ... = A157241

1, 2, 0, -8, -16, 0, 64, 128, 0, -512, ... = A088138

1, -2, -8, -8, 16, 64, 64, -128, -512, -512, ... = A138230

-3, -6, 0, 24, 48, 0, -192, -384, 0, 1536, ...

-3, 6, 24, 24, -48, -192, -192, 384, 1536, 1536, ...

9, 18, 0, -72, -144, 0, 576, 1152, 0, -4608, ...

9, -18, -72 -72, 144, 576, 576, -1152, -4608, -4608, ...

...

a(n) is the n-th term of the first column.

Successive differences of a(n) begin:

0, 1, 1, -3, -3, 9, 9, -27, -27, 81, ...

1, 0, -4, 0, 12, 0, -36, 0, 108, 0, ...

-1, -4, 4, 12, -12, -36, 36, 108, -108, -324, ...

-3, 8, 8, -24, -24, 72, 72, -216, -216, 648, ...

11, 0, -32, 0, 96, 0, -288, 0, 864, 0, ...

-11, -32, 32, 96, -96, -288, 288, 864, -864, -2592, ...

-21, 64, 64, -192, -192, 576, 576, -1728, -1728, 5184, ...

85, 0, -256, 0, 768, 0, -2304, 0, 6912, 0, ...

...

First column appears to be a subsequence of Jacobsthal numbers A001045 (the trisection A082311 is missing), second column is A104538, and third column is A137717.

a(n) = A128019(n-2) for n > 2. - Georg Fischer, Oct 23 2018

Links

• Index entries for linear recurrences with constant coefficients, signature (0, -3).

Crossrefs

Cf. A001045, A082311, A088138, A104538, A108411, A128019, A137717, A138230, A140429, A157241.

Programs

Mathematica

Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
(* or, computation from b = A157241 : *)
b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]

PARI

concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ Felix Fröhlich, Oct 23 2018

Formula

a(n) = -3*a(n-2) for n > 2.

E.g.f.: (1 - cos(sqrt(3)*x) + sqrt(3)*sin(sqrt(3)*x))/3. - Stefano Spezia, Jul 15 2024