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A291219 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^3. 49
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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                            A289342

(1 - S)^3                            A276120

(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)

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.

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 *)

PROG

(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

CROSSREFS

Cf. A000035, A290890, A291000.

Sequence in context: A192664 A122997 A146042 * A284358 A283584 A283702

Adjacent sequences:  A291216 A291217 A291218 * A291220 A291221 A291222

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 24 2017

STATUS

approved

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Last modified June 19 00:55 EDT 2019. Contains 324217 sequences. (Running on oeis4.)