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).
See A291000 for a guide to related sequences.
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7).
FORMULA
G.f.: 8*x/(1 - 2*x - 7*x^2).
a(n) = 2*a(n-1) + 7*a(n-2) for n >= 3.
a(n) = 8*A015519(n).
a(n) = sqrt(2)*((1+2*sqrt(2))^n - (1-2*sqrt(2))^n). - Colin Barker, Aug 23 2017
MATHEMATICA
z = 60; s = x/(1 - x); p = 1 - s^8;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291001 *)
LinearRecurrence[{2, 7}, {0, 8}, 41] (* G. C. Greubel, Apr 25 2023 *)
PROG
(Magma) [n le 2 select 8*(n-1) else 2*Self(n-1) +7*Self(n-2): n in [1..41]]; // G. C. Greubel, Apr 25 2023
(SageMath)
A291001=BinaryRecurrenceSequence(2, 7, 0, 8)
[A291001(n) for n in range(41)] # G. C. Greubel, Apr 25 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 22 2017
STATUS
approved