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A033113
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Base 3 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.
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13
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1, 3, 10, 30, 91, 273, 820, 2460, 7381, 22143, 66430, 199290, 597871, 1793613, 5380840, 16142520, 48427561, 145282683, 435848050, 1307544150, 3922632451, 11767897353, 35303692060, 105911076180, 317733228541, 953199685623
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OFFSET
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1,2
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COMMENTS
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Partial sums of round(3^n/4). [From Mircea Merca, Dec 28 2010]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 906
Index to sequences with linear recurrences with constant coefficients, signature (3,1,-3).
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FORMULA
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a(n) = 3*a(n-1) + a(n-2) -3*a(n-3), [From R. J. Mathar, Jun 28 2010]
G.f.: x/((1-x)*(1+x)*(1-3*x)). a(n)=2*a(n-1)+3*a(n-2)+1. Partial sums of A015518. - Paul Barry, Nov 12 2003
E.g.f.: (1/2)*exp(x)*(sinh(x))^2 - Paul Barry, Mar 12 2003
a(n)=sum{k=0..floor(n/2), C(n+2, 2k+2)*4^k }. - Paul Barry, Aug 24 2003
a(n)=sum{k=0..floor(n/2), 3^(n-2*k) }; a(n)=sum{k=0..n, sum{j=0..k, (-1)^(j+k)*3^j }}. - Paul Barry, Nov 12 2003
Convolution of A000244 and A059841 (3^n and periodic{1, 0}). a(n)=sum{k=0..n, (1+(-1)^(n-k))*3^k/2 } - Paul Barry, Jul 19 2004
a(n)=(1/8)*(9*3^(n-1)+(-1)^(n-1)-2), with n>=1 [From Paolo P. Lava, Jan 19 2009]
a(n) = round(3^(n+1)/8) = round((3^(n+1)-2)/8) = floor((3^(n+1)-1)/8) = ceil((3^(n+1)-3)/8) = round((3^(n+1)-3)/8). a(n)=a(n-2)+3^(n-1) , n>2. [From Mircea Merca, Dec 27 2010]
a(n) = floor((3^(n+1))/4) / 2, n>=1. [From Wolfdieter Lang, Apr 13 2012]
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MAPLE
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a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+1 od: seq(a[n], n=1..33); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2008]
g:=x*(1/(1-3*x)/(1-x))/(1+x): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=1..30); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 11 2009]
A033113 := proc(n) (9*3^(n-1)+(-1)^(n-1)-2)/8 ; end proc: # R. J. Mathar, Jan 08 2011
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MATHEMATICA
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Join[{a=1, b=3}, Table[c=2*b+3*a+1; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Feb 01 2011*)
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PROG
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(Pari) a(n)=if(n<0, 0, 3^n*3\8)
(MAGMA) [Round(3^(n+1)/8): n in [1..30]]; // Vincenzo Librandi, Jun 25 2011
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CROSSREFS
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Pairwise sums seem to be in A003462.
Equals A039300 - 1.
Sequence in context: A026327 A014531 A062107 * A003441 A136841 A136846
Adjacent sequences: A033110 A033111 A033112 * A033114 A033115 A033116
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Replaced duplicate of a recurrence by another recurrence - R. J. Mathar, Jun 28 2010
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STATUS
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approved
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