A255814 Numbers of words on {0,1,2,3,4,} having no isolated zeros.

1, 4, 17, 73, 313, 1341, 5745, 24613, 105449, 451773, 1935521, 8292309, 35526553, 152205613, 652091089, 2793739205, 11969154121, 51279178141, 219694231041, 941231059125, 4032495084025, 17276328107789, 74016584439345, 317107590101669

Offset: 0

Author

Milan Janjic, Mar 07 2015

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 11.
• Index entries for linear recurrences with constant coefficients, signature (5,-4,4).

Crossrefs

Cf. A255116, A255118, A254658, A254660, A255633, A255630.

Programs

Magma

I:=[1,4,17]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 26 2018

Mathematica

RecurrenceTable[{a[0] == 1, a[1] == 4,  a[2]== 17, a[n] == 5 a[n - 1] - 4 a[n - 2] + 4 a[n - 3]}, a[n], {n, 0, 23}]
LinearRecurrence[{5, -4, 4}, {1, 4, 17}, 100] (* G. C. Greubel, Jun 02 2016 *)
CoefficientList[Series[(-1 + x - x^2) / (-1 + 5 x -4 x^2 + 4 x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 26 2018 *)

Formula

a(n+3) = 5*a(n+2) - 4*a(n+1)+ 4*a(n) with n>=0, a(0) = 1, a(1) = 4, a(2) = 17.

G.f.: (-1 + x - x^2)/(-1 + 5*x - 4*x^2 + 4*x^3). - R. J. Mathar, Nov 07 2015