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A037576 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.

%I A037576 #48 Feb 16 2025 08:32:37
%S A037576 1,7,29,119,477,1911,7645,30583,122333,489335,1957341,7829367,
%T A037576 31317469,125269879,501079517,2004318071,8017272285,32069089143,
%U A037576 128276356573,513105426295,2052421705181,8209686820727,32838747282909
%N A037576 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.
%C A037576 Partial sums of A255465. - _Klaus Purath_, Mar 18 2021
%H A037576 Vincenzo Librandi, <a href="/A037576/b037576.txt">Table of n, a(n) for n = 1..1000</a>
%H A037576 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule190.html">Rule 190</a>
%H A037576 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A037576 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A037576 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,1,-4).
%F A037576 G.f.: x*(1+3*x)/((1-x)*(1-4*x)*(1+x)). - _Vincenzo Librandi_, Jun 22 2012
%F A037576 a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - _Vincenzo Librandi_, Jun 22 2012
%F A037576 a(n) = (7*4^n + 3*(-1)^n - 10)/15. - _Bruno Berselli_, Jun 22 2012, corrected by _Klaus Purath_, Mar 18 2021.
%F A037576 a(n) = floor(7*4^n/15). - _Karl V. Keller, Jr._, Mar 09 2021
%F A037576 From _Klaus Purath_, Mar 18 2021: (Start)
%F A037576 a(n) = 16*a(n-2) - 3*(-1)^n + 10, assuming that a(0) = 0.
%F A037576 a(n) = 4*a(n-1) + 2 + (-1)^n.
%F A037576 a(n) = 5*a(n-1) - 4*a(n-2) + 2*(-1)^n, n > 2. (End)
%t A037576 CoefficientList[Series[(1+3*x)/((x-1)*(4*x-1)*(1+x)),{x,0,30}],x] (*or*) LinearRecurrence[{4,1,-4},{1,7,29},40] (* _Vincenzo Librandi_, Jun 22 2012 *)
%o A037576 (Magma) I:=[1, 7, 29]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 22 2012
%o A037576 (PARI) my(x='x+O('x^99)); Vec(x*(1+3*x)/((1-x)*(1-4*x)*(1+x))) \\ _Altug Alkan_, Sep 21 2018
%o A037576 (Python) print([7*4**n//15 for n in range(1,30)]) # _Karl V. Keller, Jr._, Mar 09 2021
%Y A037576 Cf. A007090 (numbers in base 4), A037582 (decimal), A265688 (binary), A118111.
%K A037576 nonn,base,easy
%O A037576 1,2
%A A037576 _Clark Kimberling_