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%I #28 Nov 14 2021 12:12:23
%S 0,1,8,76,704,6544,60800,564928,5249024,48771328,453158912,4210527232,
%T 39122124800,363503325184,3377492099072,31381976694784,
%U 291585718747136,2709269470314496,25173184387481600,233896708743626752,2173251882598793216
%N Expansion of x/(1-8*x-12*x^2).
%C The limit of a(n+1)/a(n) is equal to 2+sqrt(7) as n approaches infinity.
%C This is the Lucas sequence U(8,-12).
%C For any three-term recurrence S(n) = S(n-1)*x1 + S(n-2)*x2*1, with S(-1) = 0 and S(0) = 1, with n-independent coefficients (like here x1=8 and x2=12) one can use the standard Morse code with a dot of length 1 standing for x1 and a dash of length 2 standing for x2. The Morse code polynomial S(x1,x2;n) is then obtained by summing over all codes of length n. E.g., S(x1,x2;3) = x1^3 + 2*x1*x2 from dot dot dot, dot dash and dash dot. Here x1=8 and x2=12 (labeled dots and dashes). For example, S(3) = 8*(8^2 + 2*12) = 704 = b(3) = a(4), because in a the offset differs from the one for S. See the Graham et al. book, on Morse code polynomials (Euler's continuants), p 302. This comment was motivated by an earlier one from the author of this sequence. - _Wolfdieter Lang_, Mar 27 2014
%C a(n-1) (for n>=1) is the number of compositions of n into 8 kinds of parts 1 and 12 kinds of parts 2. - _Joerg Arndt_, Mar 26 2014
%D R. L. Graham, D. E. Knuth, L. O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.
%H Vincenzo Librandi, <a href="/A239549/b239549.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,12).
%F a(n) = 8*a(n-1) + 12*a(n-2) for n > 1 and a(0)=0, a(1)=1.
%F G.f.: x/(1 - 8*x - 12*x^2).
%F a(n) = (1/(4*sqrt(7)))*( (4+2*sqrt(7))^n - (4-2*sqrt(7))^n ).
%F a(0) = 0 and a(n) = 2^(n-1) * A015530(n) for n > 0.
%F a(n) = A011782(n) * A015530(n) =2^(n-1)*A015530(n) for n >= 0.
%F a(n+1) = b(n) = sum(binomial(n-k, k)*8^(n-2*k)*12^k, k = 0..floor(n/2)), n>=0, b(-1) := 0. From Morse code counting, with n-2*k the number of dots and k the number of dashes for code length n. See the comment and example for b(3) = S(3) above. - _Wolfdieter Lang_, Mar 26 2014
%t CoefficientList[Series[x / (1 - 8 x - 12 x^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 28 2014 *)
%t LinearRecurrence[{8,12},{0,1},30] (* _Harvey P. Dale_, Nov 14 2021 *)
%o (Haskell)
%o a239549 n = a239549_list !! n
%o a239549_list = 0 : 1 : zipWith (+)
%o (map (* 8) $ tail a239549_list) (map (* 12) a239549_list)
%o -- _Reinhard Zumkeller_, Feb 20 2015
%Y Cf. A000079, A011782, A015530.
%K nonn,easy
%O 0,3
%A _Felix P. Muga II_, Mar 21 2014
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