A042950 Row sums of the Lucas triangle A029635.

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

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.

This sequence is d when b=[ 0,1,1,1,1,...].

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)). - Jaroslav Krizek, Aug 17 2009

Row sums of the Lucas triangle A029635. - Sergio Falcon, Mar 17 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.

Index entries for linear recurrences with constant coefficients, signature (2).

Formula

G.f.: (2-x)/(1-2*x).

a(n) = 2*a(n-1), n > 1; a(0)=2, a(1)=3.

a(n) = A003945(n), for n > 0.

From Paul Barry, Dec 06 2004: (Start)

Binomial transform of 2, 1, 2, 1, 2, 1, ... = (3+(-1)^n)/2.

a(n) = (3*2^n + 0^n)/2. (End)

a(0) = 2, a(n) = 3*2^(n-1) = 2^n + 2^(n-1) for n >= 1. - Jaroslav Krizek, Aug 17 2009

a(n) = 2^(n+1) - 2^(n-1), for n > 0. - Ilya Gutkovskiy, Aug 08 2015

Mathematica

Table[ Ceiling[3*2^(n - 1)], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
a[0] = 2; a[1] = 3; a[n_] := 2a[n - 1]; Table[a[n], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
f[s_] := Append[s, 1 + Plus @@ s]; Nest[f, {2}, 32] (* Robert G. Wilson v, Jul 08 2006 *)
CoefficientList[Series[(2 - x)/(1 - 2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 08 2006 *)

Prog

(PARI) a(n)=ceil(3*2^(n-1))
(Magma) [2] cat [2^(n+1) - 2^(n-1): n in [1..40]]; // Vincenzo Librandi, Aug 08 2015

Crossrefs

Cf. A007283.

Sequence in context: A251766 A098011 A110164 * A367222 A035055 A119559

Adjacent sequences: A042947 A042948 A042949 * A042951 A042952 A042953

Keyword

nonn,easy

Author

N. J. A. Sloane and J. H. Conway

Status

approved