login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 15 04:23 EST 2019. Contains 329991 sequences. (Running on oeis4.)