A098011 10^a(n) + 1 = A088773(n).

1, 1, 2, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 1

Author

Ray G. Opao, Sep 09 2004

Keywords

• nonn
• easy

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, 1100, 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).

Crossrefs

Cf. A119440.

Programs

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 *)

PARI

x='x+O('x^99); Vec(x*(1-x-x^3)/(1-2*x)) \\ Altug Alkan, Jul 18 2018

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

Extensions

More terms from Emeric Deutsch, May 19 2006