|
| |
|
|
A042950
|
|
The sequence d when b=[ 0,1,1,1,1,.. ].
|
|
20
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
Map a binary sequence b=[ b_1,... ] to a binary sequence c=[ c_1,... ] so that C=1/Product((1-x^i)^c_i == 1+Sum b_i*x^i mod 2.
This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C=1+Sum e_i*x^i.
Number of rises after n+1 iterations of morphism A007413.
a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). [From Jaroslav Krizek, Aug 17 2009]
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns....
|
|
|
FORMULA
|
G.f.: (2-x)/(1-2*x)
a(n)=2*a(n-1), n>1; a(0)=2, a(1)=3.
Binomial transform of 2, 1, 2, 1, 2, 1...=(3+(-1)^n)/2. a(n)=(3*2^n+0^n)/2 - Paul Barry, Dec 06 2004
a(0) = 2, a(n) = 3*2^(n-1) = 2^n + 2^(n-1) for n >= 1. [From Jaroslav Krizek, Aug 17 2009]
|
|
|
MATHEMATICA
|
Table[ Ceiling[3*2^(n - 1)], {n, 0, 32}] (* or *)
a[0] = 2; a[1] = 3; a[n_] := 2a[n - 1]; Table[a[n], {n, 0, 32}] (* or *)
f[s_] := Append[s, 1 + Plus @@ s]; Nest[f, {2}, 32] (* or *)
CoefficientList[Series[(2 - x)/(1 - 2x), {x, 0, 32}], x] (from Robert G. Wilson v, Jul 08 2006)
|
|
|
PROG
|
(PARI) a(n)=ceil(3*2^(n-1))
|
|
|
CROSSREFS
|
A003945(n)=A042950(n), n>0. Cf. A007283.
Sequence in context: A049890 * A098011 A110164 A035055 A119559 A045761
Adjacent sequences: A042947 A042948 A042949 * A042951 A042952 A042953
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane and J. H. Conway (conway(AT)math.princeton.edu)
|
|
|
STATUS
|
approved
|
| |
|
|
