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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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16
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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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LINKS
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FORMULA
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a(n) = 3*a(n-1) + a(n-2) -3*a(n-3). - 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.
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) = round(3^(n+1)/8) = floor((3^(n+1)-1)/8) = ceiling((3^(n+1)-3)/8) = round((3^(n+1)-3)/8). a(n) = a(n-2) + 3^(n-1), n > 2. - Mircea Merca, Dec 27 2010
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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); # Zerinvary Lajos, 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); # Zerinvary Lajos, Jan 11 2009
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MATHEMATICA
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Module[{nn=30, d}, d=PadRight[{}, nn, {1, 0}]; Table[FromDigits[Take[d, n], 3], {n, nn}]] (* or *) LinearRecurrence[{3, 1, -3}, {1, 3, 10}, 30] (* Harvey P. Dale, May 24 2014 *)
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PROG
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(PARI) a(n)=3^n*3\8 \\ Simplified by M. F. Hasler, Oct 06 2018
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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