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

%I A033114 #59 Dec 05 2024 13:01:12
%S A033114 1,4,17,68,273,1092,4369,17476,69905,279620,1118481,4473924,17895697,
%T A033114 71582788,286331153,1145324612,4581298449,18325193796,73300775185,
%U A033114 293203100740,1172812402961,4691249611844,18764998447377,75059993789508
%N A033114 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.
%H A033114 Vincenzo Librandi, <a href="/A033114/b033114.txt">Table of n, a(n) for n = 1..1000</a>
%H A033114 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 A033114 Madeleine Goertz and Aaron Williams, <a href="https://arxiv.org/abs/2411.19291">The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles</a>, arXiv:2411.19291 [math.CO], 2024. See p. 17.
%H A033114 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,1,-4).
%F A033114 a(n) = floor(4^(n+1)/15) = 4^(n+1)/15 - 1/6 - (-1)^n/10. - _Benoit Cloitre_, Apr 18 2003
%F A033114 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 A033114 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 A033114 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 A033114 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 A033114 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 A033114 a(n) = A117616(n)/2. - _J. M. Bergot_, Apr 22 2015
%F A033114 a(n) = A043291(n)/3; a(n+1) = 4*a(n) + A000035(n). - _Robert Israel_, Apr 22 2015
%F A033114 a(n)+a(n+1) = A002450(n+1). - _R. J. Mathar_, Feb 27 2019
%p A033114 seq(floor((4^(n+1)-1)/15),n=1..25) # _Mircea Merca_, Dec 28 2010
%t A033114 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 A033114 (Magma) [Round((8*4^n-5)/30): n in [1..30]]; // _Vincenzo Librandi_, Jun 25 2011
%Y A033114 Cf. A015521, A043291, A117616.
%K A033114 nonn,base,easy
%O A033114 1,2
%A A033114 _Clark Kimberling_