A228603 a(1) = 9, a(2) = 44, a(n) = 4*(a(n-1) + a(n-2)) (n >=3).

9, 44, 212, 1024, 4944, 23872, 115264, 556544, 2687232, 12975104, 62649344, 302497792, 1460588544, 7052345344, 34051735552, 164416323584, 793872236544, 3833154240512, 18508105908224, 89365040594944, 431492586012672, 2083430506430464, 10059692369772544

Offset: 1

Author

Emeric Deutsch, Nov 02 2013

Keywords

• nonn
• easy

Comments

a(n) = number of independent vertex subsets (i.e. the Merrifield-Simmons index) of the normal alkyl radical of n carbons (i.e. CH_3(CH_2)_{n-1}).

References

• R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 161-162.

Links

• H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20,1982, 16-21.
• Index entries for linear recurrences with constant coefficients, signature (4,4).

Crossrefs

Cf. A228602.

Programs

Maple

a := proc (n) if n = 1 then 9 elif n = 2 then 44 else 4*a(n-1)+4*a(n-2) end if end proc: seq(a(n), n = 1 .. 25);

Mathematica

LinearRecurrence[{4,4},{9,44},30] (* Harvey P. Dale, Oct 30 2016 *)

Formula

a(n) = (8 - 5*sqrt(2))*(2 - 2*sqrt(2))^(n)/8 + (8 + 5*sqrt(2))*(2 + 2*sqrt(2))^(n)/8.

G.f.: x*(9+8*x)/(1-4*x-4*x^2).

a(n) = 9*A057087(n-1)+8*A057087(n-2). - R. J. Mathar, Nov 24 2013