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A007613 a(n) = (8^n + 2*(-1)^n)/3.

%I A007613 M2129 #54 Jul 04 2025 18:42:43
%S A007613 1,2,22,170,1366,10922,87382,699050,5592406,44739242,357913942,
%T A007613 2863311530,22906492246,183251937962,1466015503702,11728124029610,
%U A007613 93824992236886,750599937895082,6004799503160662,48038396025285290,384307168202282326,3074457345618258602
%N A007613 a(n) = (8^n + 2*(-1)^n)/3.
%C A007613 Also, the cogrowth sequence of C3 X C3 = <S,T | S^3, T^3, STS^2T^2>; that is, the number of words of length 3n that reduce to the identity. - _Sean A. Irvine_, Nov 04 2024
%D A007613 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007613 Vincenzo Librandi, <a href="/A007613/b007613.txt">Table of n, a(n) for n = 0..1000</a>
%H A007613 D. S. Clark, <a href="http://www.jstor.org/stable/2691507">Proof without words</a>, Math. Mag., 63 (1990), 29.
%H A007613 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,8).
%F A007613 a(n) = A078008(3*n). - _Paul Barry_, Nov 29 2003
%F A007613 From _Paul Barry_, Mar 24 2004: (Start)
%F A007613 a(n) = (A082311(n) + (-1)^n)/2.
%F A007613 a(n) = (A001045(3*n+1) + (-1)^n)/2. (End)
%F A007613 a(n) = Sum_{k=0..n} binomial(3*n, 3*k). - _Paul Barry_, Jan 13 2005
%F A007613 a(n) = 8*a(n-1) + 6*(-1)^n. - _Paul Curtz_, Nov 19 2007
%F A007613 From _Colin Barker_, Sep 29 2014: (Start)
%F A007613 a(n) = 7*a(n-1) + 8*a(n-2).
%F A007613 G.f.: (1-5*x) / ((1+x)*(1-8*x)). (End)
%F A007613 E.g.f.: (1/3)*(exp(8*x) + 2*exp(-x)). - _G. C. Greubel_, Apr 23 2023
%t A007613 LinearRecurrence[{7,8}, {1,2}, 41] (* _G. C. Greubel_, Apr 23 2023 *)
%o A007613 (PARI) a(n)=(8^n + 2*(-1)^n)/3 \\ _Charles R Greathouse IV_, Jun 06 2011
%o A007613 (Magma) [(8^n + 2*(-1)^n)/3: n in [0..30]]; // _Vincenzo Librandi_, Aug 14 2011
%o A007613 (PARI) Vec((5*x-1)/((x+1)*(8*x-1)) + O(x^50)) \\ _Colin Barker_, Sep 29 2014
%o A007613 (SageMath) [(8^n -4*(n%2) +2)/3 for n in range(41)] # _G. C. Greubel_, Apr 23 2023
%Y A007613 Cf. A001045, A006566, A078008, A082311, A139459.
%K A007613 nonn,easy
%O A007613 0,2
%A A007613 _N. J. A. Sloane_, _Robert G. Wilson v_
%E A007613 More terms from _Colin Barker_, Sep 29 2014