

A239549


Expansion of x/(18*x12*x^2).


2



0, 1, 8, 76, 704, 6544, 60800, 564928, 5249024, 48771328, 453158912, 4210527232, 39122124800, 363503325184, 3377492099072, 31381976694784, 291585718747136, 2709269470314496, 25173184387481600, 233896708743626752, 2173251882598793216
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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 threeterm recurrence S(n) = S(n1)*x1 + S(n2)*x2*1, with S(1) = 0 and S(0) = 1, with nindependent 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(n1) (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. AddisonWesley, 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(n1) + 12*a(n2) 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  (42*sqrt(7))^n ).
a(0) = 0 and a(n) = 2^(n1) * A015530(n) for n > 0.
a(n) = A011782(n) * A015530(n) for n >= 0.
a(n+1) = b(n) = sum(binomial(nk, k)*8^(n2*k)*12^k, k = 0..floor(n/2)), n>=0, b(1) := 0. From Morse code counting, with n2*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



