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A077854 Expansion of 1/((1-x)*(1-2*x)*(1+x^2)). 10
1, 3, 6, 12, 25, 51, 102, 204, 409, 819, 1638, 3276, 6553, 13107, 26214, 52428, 104857, 209715, 419430, 838860, 1677721, 3355443, 6710886, 13421772, 26843545, 53687091, 107374182, 214748364, 429496729, 858993459, 1717986918, 3435973836, 6871947673 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Partial sums of A007910. - Mircea Merca, Dec 27 2010

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-2).

FORMULA

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.

a(n) = (1/10)*(2^(n+4) + (-1)^floor(n/2) - 2*(-1)^floor((n+1)/2) - 5).

Row sums of A130306. - Gary W. Adamson, May 20 2007

a(n) = floor(2^(n+3)/5). [Gary Detlefs, Sep 06 2010]

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]

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]

a(n) = floor(A000975(n+2)*3/5). [Armands Strazds, Oct 18 2014]

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

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

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

EXAMPLE

The sequence in hexadecimal shows a nice pattern:

1, 3, 6, c,

19, 33, 66, cc,

199, 333, 666, ccc,

1999, 3333, 6666, cccc,

19999, 33333, 66666, ccccc,

199999, 333333, 666666, cccccc,

1999999, 3333333, 6666666, ccccccc,

19999999, 33333333, 66666666, cccccccc,

... - Armands Strazds, Oct 09 2014

MAPLE

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;

seq(iquo(2^n, 5), n=3..35); # Zerinvary Lajos, Apr 20 2008

MATHEMATICA

CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 + x^2)), {x, 0, 32}], x] (* Michael De Vlieger, Mar 29 2016 *)

LinearRecurrence[{3, -3, 3, -2}, {1, 3, 6, 12}, 40] (* Harvey P. Dale, Feb 06 2019 *)

PROG

(MAGMA) [Round((2^(n+4)-5)/10): n in [0..40]]; // Vincenzo Librandi, Jun 25 2011

(PARI) a(n)=(16<<n)\10 \\ Charles R Greathouse IV, Sep 23 2012

(PARI) Vec(1/(1-3*x+3*x^2-3*x^3+2*x^4)+O(x^99)) \\ Derek Orr, Oct 26 2014

(Haskell)

import Data.Bits (xor)

a077854 n = a077854_list !! n

a077854_list = scanl1 xor $ tail a000975_list :: [Integer]

-- Reinhard Zumkeller, Jan 04 2013

CROSSREFS

Equals A007909(n+3) - [n congruent 2, 3 mod 4].

Cf. A130306.

Cf. A043291 (subsequence); A000975.

Sequence in context: A088970 A068425 A136444 * A265700 A293313 A267539

Adjacent sequences:  A077851 A077852 A077853 * A077855 A077856 A077857

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 17 2002

STATUS

approved

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Last modified June 17 17:12 EDT 2019. Contains 324196 sequences. (Running on oeis4.)