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A289783
p-INVERT of the (3^n), where p(S) = 1 - S - S^2.
2
1, 5, 24, 113, 527, 2446, 11325, 52369, 242008, 1117997, 5163891, 23849270, 110142089, 508652653, 2349005592, 10847859961, 50095958215, 231345247934, 1068361195173, 4933730638937, 22784141325656, 105217952251285, 485900111176779, 2243903303473318
OFFSET
0,2
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).
FORMULA
G.f.: (1 - 2 x)/(1 - 7 x + 11 x^2).
a(n) = 7*a(n-1) - 11*a(n-2).
a(n) = (2^(-n-1)*((7-sqrt(5))^(n+1)*(-4+sqrt(5)) + (4+sqrt(5))*(7+sqrt(5))^(n+1))) / (11*sqrt(5)). - Colin Barker, Aug 11 2017
a(n) = A099453(n)-2*A099453(n-1). - R. J. Mathar, Jul 08 2022
MATHEMATICA
z = 60; s = x/(1 - 3*x); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000244 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289783 *)
PROG
(PARI) Vec(x*(1 - 2*x) / (1 - 7*x + 11*x^2) + O(x^30)) \\ Colin Barker, Aug 11 2017
CROSSREFS
Sequence in context: A272257 A347029 A141223 * A140766 A026388 A242509
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 10 2017
STATUS
approved