A289925 p-INVERT of the lower Wythoff sequence (A000201), where p(S) = 1 - S - S^2.

1, 5, 19, 72, 265, 979, 3618, 13374, 49447, 182807, 675843, 2498594, 9237316, 34150422, 126254366, 466763346, 1725627604, 6379658213, 23585644300, 87196304028, 322365390600, 1191787269208, 4406046481612, 16289186920873, 60221246337260, 222638399818776

Offset: 0

Author

Clark Kimberling, Aug 14 2017

Keywords

• nonn
• easy

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 A289780 for a guide to related sequences.

Crossrefs

Cf. A000201, A289926.

Programs

Mathematica

z = 60; r = GoldenRatio; s = Sum[Floor[k*r] x^k, {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000201 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289925 *)