A228604 The Merrifield-Simmons index of the ortho-polyphenylene chain of length n.

1, 18, 299, 4932, 81301, 1340118, 22089599, 364109832, 6001737001, 98928520218, 1630669938899, 26878845894732, 443052477632701, 7302973450020318, 120377210159548199, 1984215446621359632, 32706447785195768401, 539110673967989840418, 8886330936793922917499

Offset

0,2

Comments

The Merrifield-Simmons index of a graph is the number of its independent vertex subsets.

References

R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989.

Links

Table of n, a(n) for n=0..18.

T. Doslic, M. S. Litz, Matchings and independent sets in polyphenylene chains, MATCH, Commun. Math. Comput. Chem., 67, 2012, 313-330.

Index entries for linear recurrences with constant coefficients, signature (18,-25).

Formula

a(n) = ((9 + 2*sqrt(14))^(n+1) - (9 - 2*sqrt(14))^(n+1))/(4*sqrt(14)).

G.f. = 1/(1 - 18*x + 25*x^2).

a(n) = 18*a(n-1) - 25*a(n-2); a(0)=1, a(1)=18. - Harvey P. Dale, Nov 06 2014

Maple

gser := series(1/(25*x^2-18*x+1), x = 0, 22): seq(coeff(gser, x, n), n = 0 .. 20);

Mathematica

CoefficientList[Series[1/(1-18x+25x^2), {x, 0, 20}], x] (* Harvey P. Dale, Nov 06 2014 *)
(* Alternative: *)
LinearRecurrence[ {18, -25}, {1, 18}, 20] (* Harvey P. Dale, Nov 06 2014 *)

Prog

(PARI) a(n)=([0, 1; -25, 18]^n*[1; 18])[1, 1] \\ Charles R Greathouse IV, May 30 2026

Crossrefs

Cf. A228605, A228606.

Sequence in context: A383172 A166927 A113367 * A228606 A228605 A386011

Adjacent sequences: A228601 A228602 A228603 * A228605 A228606 A228607

Keyword

nonn,easy,changed

Author

Emeric Deutsch, Nov 23 2013

Status

approved