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A033114 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,0. 8
1, 4, 17, 68, 273, 1092, 4369, 17476, 69905, 279620, 1118481, 4473924, 17895697, 71582788, 286331153, 1145324612, 4581298449, 18325193796, 73300775185, 293203100740, 1172812402961, 4691249611844, 18764998447377, 75059993789508 (list; graph; refs; listen; history; text; internal format)
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: A081113 A114587 A268431 * A096881 A033122 A330246

Adjacent sequences:  A033111 A033112 A033113 * A033115 A033116 A033117

KEYWORD

nonn,base,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 17 21:31 EDT 2021. Contains 343070 sequences. (Running on oeis4.)