A043291 Every run length in base 2 is 2.

3, 12, 51, 204, 819, 3276, 13107, 52428, 209715, 838860, 3355443, 13421772, 53687091, 214748364, 858993459, 3435973836, 13743895347, 54975581388, 219902325555, 879609302220, 3518437208883, 14073748835532, 56294995342131, 225179981368524, 900719925474099

Offset

1,1

Comments

a(n) is the number whose binary representation is A153435(n). - Omar E. Pol, Jan 18 2009

See A033001 and following for the analog in other bases and the variant with runs of length >= 2. - M. F. Hasler, Feb 01 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Formula

a(n) = 4*a(n-1)+a(n-2)-4*a(n-3), n>3. - John W. Layman, Feb 01 2000

a(n) = floor(4^(n+1)/5). - Mircea Merca, Dec 26 2010

G.f.: 3*x / ( (x-1)*(4*x-1)*(1+x) ). - Joerg Arndt, Jan 08 2011

a(n) = 3*A033114(n). - R. J. Mathar, Jan 08 2011

Maple

seq(floor(4^(n+1)/5), n=1..25); # Mircea Merca, Dec 26 2010

Mathematica

f[n_] := Floor[4^(n + 1)/5]; Array[f, 23] (* or *)
a[1] = 3; a[2] = 12; a[3] = 51; a[n_] := a[n] = 4 a[n - 1] + a[n - 2] - 4 a[n - 3]; Array[a, 23] (* or *)
3 LinearRecurrence[{4, 1, -4}, {1, 4, 17}, 23] (* Robert G. Wilson v, Jul 01 2014 *)

Prog

(Magma) [Floor(4^(n+1)/5): n in [1..30]]; // Vincenzo Librandi, Jun 26 2011
(PARI) A043291 = n->4^(n+1)\5 \\ M. F. Hasler, Feb 01 2014
(Python)
def a(n): return int(''.join([['11', '00'][i%2] for i in range(n)]), 2)
print([a(n) for n in range(1, 26)]) # Michael S. Branicky, Mar 12 2021

Crossrefs

Cf. A153435 (binary).

Bisections: A108020, A182512. Bisection of A077854.

Cf. A000975, A033001, A033114.

Sequence in context: A242155 A009024 A265083 * A135343 A083314 A155179

Adjacent sequences: A043288 A043289 A043290 * A043292 A043293 A043294

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved