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A239549 Expansion of x/(1-8*x-12*x^2). 2
0, 1, 8, 76, 704, 6544, 60800, 564928, 5249024, 48771328, 453158912, 4210527232, 39122124800, 363503325184, 3377492099072, 31381976694784, 291585718747136, 2709269470314496, 25173184387481600, 233896708743626752, 2173251882598793216 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The limit of a(n+1)/a(n) is equal to 2+sqrt(7) as n approaches infinity.

This is the Lucas sequence U(8,-12).

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

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

REFERENCES

R. L. Graham, D. E. Knuth, L. O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,12).

FORMULA

a(n) = 8*a(n-1) + 12*a(n-2) for n > 1 and a(0)=0, a(1)=1.

G.f.: x/(1 - 8*x - 12*x^2).

a(n) = (1/(4*sqrt(7)))*( (4+2*sqrt(7))^n - (4-2*sqrt(7))^n ).

a(0) = 0 and a(n) = 2^(n-1) * A015530(n) for n > 0.

a(n) = A011782(n) * A015530(n) for n >= 0.

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

MATHEMATICA

CoefficientList[Series[x / (1 - 8 x - 12 x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 28 2014 *)

PROG

(Haskell)

a239549 n = a239549_list !! n

a239549_list = 0 : 1 : zipWith (+)

               (map (* 8) $ tail a239549_list) (map (* 12) a239549_list)

-- Reinhard Zumkeller, Feb 20 2015

CROSSREFS

Cf. A000079, A011782, A015530.

Sequence in context: A302814 A088376 A096293 * A247828 A303736 A083234

Adjacent sequences:  A239546 A239547 A239548 * A239550 A239551 A239552

KEYWORD

nonn,easy

AUTHOR

Felix P. Muga II, Mar 21 2014

STATUS

approved

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Last modified August 5 07:25 EDT 2020. Contains 336209 sequences. (Running on oeis4.)