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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                            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) 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 January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)