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A133467 a(n) = a(n-1) + 6*a(n-2) for n >= 2, a(0)=1, a(1)=2.

%I A133467 #45 Jun 07 2025 20:04:22
%S A133467 1,2,8,20,68,188,596,1724,5300,15644,47444,141308,425972,1273820,
%T A133467 3829652,11472572,34450484,103285916,309988820,929704316,2789637236,
%U A133467 8367863132,25105686548,75312865340,225946984628,677824176668,2033506084436,6100451144444,18301487651060
%N A133467 a(n) = a(n-1) + 6*a(n-2) for n >= 2, a(0)=1, a(1)=2.
%C A133467 From _Sean A. Irvine_, Jun 07 2025: (Start)
%C A133467 The number of walks of length n starting at vertex 0 (or, by symmetry, vertex 2 or 3) in the graph K_{1,1,3}:
%C A133467     1---2
%C A133467    /|\ /
%C A133467   0 | X
%C A133467    \|/ \
%C A133467     4---3. (End)
%H A133467 Harvey P. Dale, <a href="/A133467/b133467.txt">Table of n, a(n) for n = 0..1000</a>
%H A133467 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A133467 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,6).
%F A133467 G.f.: (1+x)/((1+2*x)*(1-3*x)).
%F A133467 a(n) = Sum_{k=0..n+1} A122950(n+1,k)*5^(n+1-k). - _Philippe Deléham_, Jan 08 2008
%F A133467 a(n) = (4 * 3^n + (-2)^n) / 5. - _Werner Schulte_, Dec 18 2024
%e A133467 a(2)=8 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3. - _Sean A. Irvine_, Jun 07 2025
%p A133467 Digits := 50:
%p A133467 for n from 0 to 40 do round(.8*3^n+.2*(-2)^n) end do;
%p A133467 # _Matt C. Anderson_, Jul 18 2017
%t A133467 LinearRecurrence[{1,6},{1,2},30] (* _Harvey P. Dale_, Apr 05 2014 *)
%o A133467 (Sage) from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,1,6, lambda n: 0); [next(it) for i in range(0,29)] # _Zerinvary Lajos_, Jul 03 2008
%K A133467 easy,nonn
%O A133467 0,2
%A A133467 _Philippe Deléham_, Jan 03 2008