OFFSET
0,2
COMMENTS
a(n) is the number of strings of length n+1 defined on {0,1,2} such that both 0 and 1 appear at least once and the combined number of 0's and 1's is even. - Enrique Navarrete, Mar 16 2026
REFERENCES
J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..2000
J. Propp, Publications and Preprints
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,6).
FORMULA
a(n) = A092437(n, n+1).
a(n) = A046717(n+1)-2^(n+1)+1.
a(n) = (3^(n+1)+(-1)^(n+1))/2-2^(n+1)+1.
From R. J. Mathar, Apr 21 2010: (Start)
a(n) = +5*a(n-1) -5*a(n-2) -5*a(n-3) +6*a(n-4) = 2*A140420(n).
G.f.: -2*x*(1-2*x+3*x^2) / ( (x-1)*(3*x-1)*(2*x-1)*(1+x) ). (End)
E.g.f.: (1/2)*exp(-x)*(3*exp(4*x) - 4*exp(3*x) + 2*exp(2*x) - 1). - Enrique Navarrete, Mar 16 2026
EXAMPLE
a(3) = (3^4+(-1)^4)/2-2^4+1 = 26.
MATHEMATICA
LinearRecurrence[{5, -5, -5, 6}, {0, 2, 6, 26}, 30] (* Paolo Xausa, Mar 23 2026 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004
EXTENSIONS
More terms from Paolo Xausa, Mar 23 2026
STATUS
approved