A291228 -id:A291228 - cp's OEIS frontend

A291257 a(n) = (1/2)*A291228(n).

1, 3, 9, 28, 85, 261, 797, 2440, 7461, 22827, 69821, 213588, 653345, 1998573, 6113529, 18701072, 57205769, 174990195, 535287793, 1637423756, 5008812525, 15321754293, 46868623381, 143369215128, 438560602669, 1341539064795, 4103713486629, 12553092811972

Offset: 0

Clark Kimberling, Aug 25 2017

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).

See A291219 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 4, -2, -1)

Cf. A000035, A291219, A291228.

z = 60; s = x/(1 - x^2); p = 1 - 2 s - 2 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291228 *)
u/2         (* A291257 *)

G.f.: -((2 (-1 - x + x^2))/(1 - 2 x - 4 x^2 + 2 x^3 + x^4)).
a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) - a(n-4) for n >= 5.
a(n) = (1/2)*A291228(n) for n >= 0.

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

Original entry on oeis.org

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

Clark Kimberling, Aug 24 2017

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

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)

Cf. A000035, A290890, A291000.

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

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 *)

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.

cp's OEIS Frontend

A291257 a(n) = (1/2)*A291228(n).

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