A120925 Number of ternary words on {0,1,2} having no isolated 0's.
1, 2, 5, 13, 33, 83, 209, 527, 1329, 3351, 8449, 21303, 53713, 135431, 341473, 860983, 2170865, 5473575, 13800961, 34797463, 87737617, 221219847, 557779233, 1406373239, 3546000945, 8940814823, 22543189057, 56839939415, 143315069777
Offset: 0
Examples
a(2)=5 because we have 00,11,12,21 and 22.
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..2490
- 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.
- Maksym Druchok, Volodymyr Krasnov, Taras Krokhmalskii, and Oleg Derzhko, One-dimensionally confined ammonia molecules: A theoretical study, arXiv:2307.06186 [cond-mat.stat-mech], 2023. See p. 5.
- Index entries for linear recurrences with constant coefficients, signature (3,-2,2).
Programs
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Maple
a[0]:=1:a[1]:=2:a[2]:=5: for n from 3 to 32 do a[n]:=3*a[n-1]-2*a[n-2]+2*a[n-3] od: seq(a[n],n=0..32);
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Mathematica
nn=20;a=x^2/(1-x);CoefficientList[Series[(a+1)/(1-(2x a)/(1-2x))/(1-2x),{x,0,nn}],x] (* Geoffrey Critzer, Jan 13 2013 *) LinearRecurrence[{3,-2,2},{1,2,5},30] (* Harvey P. Dale, Nov 16 2024 *)
Formula
a(n) = 3a(n-1)-2a(n-2)+2a(n-3); a(0)=1, a(1)=2,a(2)=5.
G.f.: (1-z+z^2)/(1-3z+2z^2-2z^3).
Comments