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URL: https://oeis.org/A042950

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A042950
Row sums of the Lucas triangle A029635.
31
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
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
E.g.f.: (3*exp(2*x) + 1)/2. - G. C. Greubel, Jun 06 2025
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
(Python)
def A042950(n): return (3*2**n + int(n==0))//2 # G. C. Greubel, Jun 06 2025
CROSSREFS
Sequence in context: A251745 A251752 A251766 * A110164 A098011 A367222
KEYWORD
nonn,easy
STATUS
approved