A120926 Number of isolated 0's in all ternary words of length n on {0,1,2}.

1, 4, 16, 60, 216, 756, 2592, 8748, 29160, 96228, 314928, 1023516, 3306744, 10628820, 34012224, 108413964, 344373768, 1090516932, 3443737680, 10847773692, 34093003032, 106928054964, 334731302496, 1046035320300, 3263630199336, 10167463313316, 31632108085872

Offset

1,2

Comments

This is essentially the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - 2 S); see A291000. - Clark Kimberling, Aug 24 2017

Links

Table of n, a(n) for n=1..27.

Index entries for linear recurrences with constant coefficients, signature (6,-9).

Formula

a(n) = (4/27)*(n+1)*3^n for n >= 2.

G.f.: z*(1-z)^2/(1-3*z)^2.

a(n) = Sum_{k=0..ceiling(n/2)} k*A120924(n,k).

From Enrique Navarrete, Nov 24 2025: (Start)

First differences of A081038; second differences of A027471.

E.g.f.: (4/27)*(exp(3*x)*(3*x + 1) + (3/4)*x - 1). (End)

Example

a(2) = 4 because in the 9 ternary words of length 2, namely 00, 01, 02, 10, 11, 12, 20, 21 and 22, we have altogether 4 isolated 0's.

Maple

1, seq(4*(n+1)*3^n/27, n=2..28);

Mathematica

LinearRecurrence[{6, -9}, {1, 4, 16}, 30] (* Harvey P. Dale, Dec 08 2025 *)

Crossrefs

Cf. A120924, A291000.

Cf. A027471, A081038.

Sequence in context: A119827 A089883 A089932 * A255303 A273347 A268939

Adjacent sequences: A120923 A120924 A120925 * A120927 A120928 A120929

Keyword

nonn,easy

Author

Emeric Deutsch, Jul 16 2006

Status

approved