A033114 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.

1, 4, 17, 68, 273, 1092, 4369, 17476, 69905, 279620, 1118481, 4473924, 17895697, 71582788, 286331153, 1145324612, 4581298449, 18325193796, 73300775185, 293203100740, 1172812402961, 4691249611844, 18764998447377, 75059993789508

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.

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

Formula

a(n) = floor(4^(n+1)/15) = 4^(n+1)/15 - 1/6 - (-1)^n/10. - Benoit Cloitre, Apr 18 2003

G.f.: 1/((1-x)*(1+x)*(1-4*x)); a(n) = 3*a(n-1) + 4*a(n-2)+1. Partial sum of A015521. - Paul Barry, Nov 12 2003

a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k); a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^(j+k)*4^j. - Paul Barry, Nov 12 2003

Convolution of A000302 and A059841 (4^n and periodic{1, 0}). a(n) = Sum_{k=0..n} (1 + (-1)^(n-k))*4^k/2. - Paul Barry, Jul 19 2004

a(n) = Sum_{k=0..n} (-1)^(n-k)*(J(2*k+1)-1)/2, J(n)=A001045(n). - Paul Barry, Mar 06 2008

a(n) = round((8*4^n-5)/30) = ceiling((4*4^n-4)/15) = round((4*4^n-4)/15); a(n) = a(n-2) + 4^(n-1), n > 1. - Mircea Merca, Dec 28 2010

a(n) = A117616(n)/2. - J. M. Bergot, Apr 22 2015

a(n) = A043291(n)/3; a(n+1) = 4*a(n) + A000035(n). - Robert Israel, Apr 22 2015

a(n)+a(n+1) = A002450(n+1). - R. J. Mathar, Feb 27 2019

Maple

seq(floor((4^(n+1)-1)/15), n=1..25) # Mircea Merca, Dec 28 2010

Mathematica

Join[{a=1, b=4}, Table[c=3*b+4*a+1; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)

Prog

(Magma) [Round((8*4^n-5)/30): n in [1..30]]; // Vincenzo Librandi, Jun 25 2011

Crossrefs

Cf. A015521, A043291, A117616.

Sequence in context: A368429 A114587 A268431 * A096881 A363588 A344217

Adjacent sequences: A033111 A033112 A033113 * A033115 A033116 A033117

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved