%I #55 Sep 08 2022 08:44:51
%S 1,4,17,68,273,1092,4369,17476,69905,279620,1118481,4473924,17895697,
%T 71582788,286331153,1145324612,4581298449,18325193796,73300775185,
%U 293203100740,1172812402961,4691249611844,18764998447377,75059993789508
%N Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.
%H Vincenzo Librandi, <a href="/A033114/b033114.txt">Table of n, a(n) for n = 1..1000</a>
%H Thomas Baruchel, <a href="https://arxiv.org/abs/1908.02250">Properties of the cumulated deficient binary digit sum</a>, arXiv:1908.02250 [math.NT], 2019.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,1,-4).
%F a(n) = floor(4^(n+1)/15) = 4^(n+1)/15 - 1/6 - (-1)^n/10. - _Benoit Cloitre_, Apr 18 2003
%F 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
%F 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
%F 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
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*(J(2*k+1)-1)/2, J(n)=A001045(n). - _Paul Barry_, Mar 06 2008
%F 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
%F a(n) = A117616(n)/2. - _J. M. Bergot_, Apr 22 2015
%F a(n) = A043291(n)/3; a(n+1) = 4*a(n) + A000035(n). - _Robert Israel_, Apr 22 2015
%F a(n)+a(n+1) = A002450(n+1). - _R. J. Mathar_, Feb 27 2019
%p seq(floor((4^(n+1)-1)/15),n=1..25) # _Mircea Merca_, Dec 28 2010
%t 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 *)
%o (Magma) [Round((8*4^n-5)/30): n in [1..30]]; // _Vincenzo Librandi_, Jun 25 2011
%Y Cf. A015521, A043291, A117616.
%K nonn,base,easy
%O 1,2
%A _Clark Kimberling_