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

%I A228603 #15 Jan 05 2025 19:51:40
%S A228603 9,44,212,1024,4944,23872,115264,556544,2687232,12975104,62649344,
%T A228603 302497792,1460588544,7052345344,34051735552,164416323584,
%U A228603 793872236544,3833154240512,18508105908224,89365040594944,431492586012672,2083430506430464,10059692369772544
%N A228603 a(1) = 9, a(2) = 44, a(n) = 4*(a(n-1) + a(n-2)) (n >=3).
%C A228603 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}).
%D A228603 R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 161-162.
%H A228603 H. Prodinger and R. F. Tichy, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-1/prodinger.pdf">Fibonacci numbers of graphs</a>, Fibonacci Quarterly, 20,1982, 16-21.
%H A228603 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4).
%F A228603 a(n) = (8 - 5*sqrt(2))*(2 - 2*sqrt(2))^(n)/8 + (8 + 5*sqrt(2))*(2 + 2*sqrt(2))^(n)/8.
%F A228603 G.f.: x*(9+8*x)/(1-4*x-4*x^2).
%F A228603 a(n) = 9*A057087(n-1)+8*A057087(n-2). - _R. J. Mathar_, Nov 24 2013
%p A228603 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);
%t A228603 LinearRecurrence[{4,4},{9,44},30] (* _Harvey P. Dale_, Oct 30 2016 *)
%Y A228603 Cf. A228602.
%K A228603 nonn,easy
%O A228603 1,1
%A A228603 _Emeric Deutsch_, Nov 02 2013