A291219 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^3.

1, 1, 3, 5, 11, 21, 42, 83, 163, 323, 635, 1255, 2473, 4880, 9625, 18985, 37451, 73869, 145715, 287421, 566954, 1118331, 2205947, 4351307, 8583091, 16930447, 33395857, 65874464, 129939569, 256310161, 505580371, 997274197, 1967156763, 3880282533, 7653987242

Offset: 0

Author

Clark Kimberling, Aug 24 2017

Keywords

• nonn
• easy

Comments

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

In the following guide to p-INVERT sequences using s = (1,0,1,0,1,...) = A000035, in some cases t(1,0,1,0,1,...) is a shifted version of the indicated sequence.

p(S) t(1,0,1,0,1,...)

1 - S A000045 (Fibonacci numbers)

1 - S^2 A147600

1 - S^3 A291217

1 - S^5 A291218

1 - S - S^2 A289846

1 - S - S^3 A291219

1 - S - S^4 A291220

1 - S^3- S^6 A291221

1 - S^2- S^3 A291222

1 - S^3- S^4 A291223

1 - 2S A052542

1 - 3S A006190

(1 - S)^2 A239342

(1 - S)^3 A276129

(1 - S)^4 A291224

(1 - S)^5 A291225

(1 - S)^6 A291226

1 - S - 2 S^2 A291227

1 - 2 S - 2 S^2 A291228

1 - 3 S - 2 S^2 A060801

(1 - S)(1 - 2 S) A291229

(1 - S)(1 - 2 S)(1 - 3 S) A291230

(1 - S)(1 - 2 S)(1 - 3 S)( 1 - 4 S) A291231

(1 - 2 S)^2 A291264

(1 - 3 S)^2 A291232

1 - S - S^2 - S^3 A291233

1 - S - S^2 - S^3 - S^4 A291234

1 - S - S^2 - S^3 - S^4 - S^5 A291235

(1 - S)(1 - 3 S) A291236

(1 - S)(1 - 2S)( 1 - 4S) A291237

(1 - S)^2 (1 - 2S) A291238

(1 - S^2) (1 - 2S) A291239

(1 - S^3)^2 A291240

1 - S - S^2 + S^3 A291241

1 - 2 S - S^2 + S^3 A291242

1 - 3 S + S^2 A291243

1 - 4 S + S^2 A291244

1 - 5 S + S^2 A291245

1 - 6 S + S^2 A291246

1 - S - S^2 - S^3 + S^4 A291247

1 - S - S^2 - S^3 - S^4 + S^5 A291248

1 - S - S^2 - S^3 + S^4 + S^5 A291249

1 - S - 2 S^2 + 2 S^3 A291250

1 - 3 S^2 + 2 S^3 A291251 (includes negative terms)

(1 - S^3)^3 A291252

(1 - S - S^2)^2 A291253

(1 - 2 S - S^2)^2 A291254

(1 - S - 2 S^2)^2 A291255

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-3,1,1)

Crossrefs

Cf. A000035, A290890, A291000.

Programs

Magma

I:=[1,1,3,5,11,21]; [n le 6 select I[n] else Self(n-1)+3*Self(n-2)-Self(n-3)-3*Self(n-4)+Self(n-5)+Self(n-6): n in [1..45]]; // Vincenzo Librandi, Aug 25 2017

Mathematica

z = 60; s = x/(1 - x^2); p = 1 - s - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291219 *)
LinearRecurrence[{1, 3, -1, -3, 1, 1}, {1, 1, 3, 5, 11, 21}, 50] (* Vincenzo Librandi, Aug 25 2017 *)

Formula

G.f.: -(1 - x^2 + x^4)/(-1 + x + 3*x^2 - x^3 - 3*x^4 + x^5 + x^6).

a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 3*a(n-4) + a(n-5) + a(n-6) for n >= 7.