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A077854 Expansion of 1/((1-x)*(1-2*x)*(1+x^2)).

%I A077854 #112 Sep 08 2022 08:45:07
%S A077854 1,3,6,12,25,51,102,204,409,819,1638,3276,6553,13107,26214,52428,
%T A077854 104857,209715,419430,838860,1677721,3355443,6710886,13421772,
%U A077854 26843545,53687091,107374182,214748364,429496729,858993459,1717986918,3435973836,6871947673
%N A077854 Expansion of 1/((1-x)*(1-2*x)*(1+x^2)).
%C A077854 Partial sums of A007910. - _Mircea Merca_, Dec 27 2010
%C A077854 This is the decimal representation of the middle column of "Rule 54" elementary cellular automaton. - _Karl V. Keller, Jr._, Sep 26 2021
%C A077854 This same sequence (except that the offset is changed to 4) is 2^n with the final digit chopped off. - _J. Lowell_, May 11 2022
%H A077854 Vincenzo Librandi, <a href="/A077854/b077854.txt">Table of n, a(n) for n = 0..1000</a>
%H A077854 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,-2).
%F A077854 a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 2*a(n-4), with initial values a(0) = 1, a(1) = 3, a(2) = 6, a(3) = 12.
%F A077854 a(n) = (1/10)*(2^(n+4) + (-1)^floor(n/2) - 2*(-1)^floor((n+1)/2) - 5).
%F A077854 Row sums of A130306. - _Gary W. Adamson_, May 20 2007
%F A077854 a(n) = floor(2^(n+3)/5). - _Gary Detlefs_, Sep 06 2010
%F A077854 a(n) = round((2^(n+4)-5)/10) = floor((2^(n+3)-1)/5) = ceiling((2^(n+3)-4)/5) = round((2^(n+3)-2)/5); a(n) = a(n-4) + 3*2^(n-1), n > 3. - _Mircea Merca_, Dec 27 2010
%F A077854 a(n) = 2^(n+1) - 1 - a(n-2); a(n) = a(n-1)/2 for n == 2, 3 (mod 4); a(n) = (a(n-1)-1)/2 for n == 0, 1 (mod 4). - _Arie Bos_, Apr 06 2013
%F A077854 a(n) = floor(A000975(n+2)*3/5). - _Armands Strazds_, Oct 18 2014
%F A077854 a(n) = Sum_{k=1..n+3} floor(1 + sin(k*Pi/2 + 3*Pi/4))*2^(n-k+3). - _Andres Cicuttin_, Mar 28 2016
%F A077854 a(n) = (-15 + 3*2^(3+n) + 2^(1 + n - 4*floor((1+n)/4)) + 2^(2 + n - 4*floor((2+n)/4)))/15. - _Andres Cicuttin_, Mar 28 2016
%F A077854 a(n) = (16*2^n+(-1)^((2*n-1+(-1)^n)/4)-2*(-1)^((2*n+1-(-1)^n)/4)-5)/10. - _Wesley Ivan Hurt_, Apr 01 2016
%e A077854 The sequence in hexadecimal shows the pattern
%e A077854 1, 3, 6, c,
%e A077854 19, 33, 66, cc,
%e A077854 199, 333, 666, ccc,
%e A077854 1999, 3333, 6666, cccc,
%e A077854 19999, 33333, 66666, ccccc,
%e A077854 199999, 333333, 666666, cccccc,
%e A077854 1999999, 3333333, 6666666, ccccccc,
%e A077854 19999999, 33333333, 66666666, cccccccc,
%e A077854 ... - _Armands Strazds_, Oct 09 2014
%p A077854 a := proc(n) option remember; if n=0 then RETURN(1); fi; if n=1 then RETURN(3); fi; if n=2 then RETURN(6); fi; if n=3 then RETURN(12); fi; 3*a(n-1)-3*a(n-2)+3*a(n-3)-2*a(n-4); end;
%p A077854 seq(iquo(2^n,5),n=3..35); # _Zerinvary Lajos_, Apr 20 2008
%t A077854 CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 + x^2)), {x, 0, 32}], x] (* _Michael De Vlieger_, Mar 29 2016 *)
%t A077854 LinearRecurrence[{3,-3,3,-2},{1,3,6,12},40] (* _Harvey P. Dale_, Feb 06 2019 *)
%o A077854 (Magma) [Round((2^(n+4)-5)/10): n in [0..40]]; // _Vincenzo Librandi_, Jun 25 2011
%o A077854 (PARI) a(n)=(16<<n)\10 \\ _Charles R Greathouse IV_, Sep 23 2012
%o A077854 (PARI) Vec(1/(1-3*x+3*x^2-3*x^3+2*x^4)+O(x^99)) \\ _Derek Orr_, Oct 26 2014
%o A077854 (Haskell)
%o A077854 import Data.Bits (xor)
%o A077854 a077854 n = a077854_list !! n
%o A077854 a077854_list = scanl1 xor $ tail a000975_list :: [Integer]
%o A077854 -- _Reinhard Zumkeller_, Jan 04 2013
%o A077854 (Python) print([2**(n+3)//5 for n in range(50)]) # _Karl V. Keller, Jr._, Sep 26 2021
%Y A077854 Equals A007909(n+3) - [n congruent 2, 3 mod 4].
%Y A077854 Cf. A130306, A043291 (subsequence); A000975, A007910, A133872, A259661 (binary).
%K A077854 nonn,easy
%O A077854 0,2
%A A077854 _N. J. A. Sloane_, Nov 17 2002