OFFSET
1,3
COMMENTS
Starting from the 4th term, every succeeding term is twice the preceding term. I.e., a(n+1) = 2a(n).
Number of binary words of length n-2 that do not start with 01 (n>=2). Example: a(5)=6 because we have 000,001,100,101,110 and 111. Except for the initial term, column 0 of A119440. - Emeric Deutsch, May 19 2006
a(n) written in base 2: a(1) = 1, a(2) = 1, a(3) = 10, a(n) for n >= 4: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-4) times 0 (see A003953(n-3). - Jaroslav Krizek, Aug 17 2009
a(n) for n > 1 are the values used in the variant of the game 2048 called "threes". - Michael De Vlieger, Jul 18 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Index entries for linear recurrences with constant coefficients, signature (2).
FORMULA
G.f.: x*(1 - x - x^3)/(1 - 2*x). - Paul Barry, Feb 17 2005
a(n) = 3*2^(n-4) for n>3; a(1)=a(2)=1, a(3)=2. - Emeric Deutsch, May 19 2006
a(n) = 2^(n-4) + 2^(n-3) for n > 3. - Jaroslav Krizek, Aug 17 2009
a(1) = 1, a(2) = 1, a(3) = 2, for n > 3: a(n) = Sum_{i = 2..n-1} a(i). - Jaroslav Krizek, Nov 16 2009 [Corrected by Petros Hadjicostas, Nov 16 2019]
a(n) = A042950(n-3). - Philippe Deléham, Oct 17 2011
a(n) = ceiling(2^{n-2}) - floor(2^{n-4}). - Martin Grymel, Oct 17 2012
MAPLE
a:=proc(n) if n=1 or n=2 then 1 elif n=3 then 2 else 3*2^(n-4) fi end: seq(a(n), n=1..37); # Emeric Deutsch, May 19 2006
MATHEMATICA
Table[ Ceiling[3*2^(n - 4)], {n, 34}] (* or *)
Rest@CoefficientList[Series[x(1 - x - x^3)/(1 - 2x), {x, 0, 33}], x] (* Robert G. Wilson v, Jul 08 2006 *)
Table[Ceiling[2^{n-2}]-Floor[2^{n-4}], {n, 1, 10}] (* Martin Grymel, Oct 17 2012 *)
PROG
(PARI) x='x+O('x^99); Vec(x*(1-x-x^3)/(1-2*x)) \\ Altug Alkan, Jul 18 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ray G. Opao, Sep 09 2004
EXTENSIONS
More terms from Emeric Deutsch, May 19 2006
STATUS
approved