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A097137
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Convolution of 3^n and floor(n/2).
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3
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0, 0, 1, 4, 14, 44, 135, 408, 1228, 3688, 11069, 33212, 99642, 298932, 896803, 2690416, 8071256, 24213776, 72641337, 217924020, 653772070, 1961316220, 5883948671, 17651846024, 52955538084, 158866614264, 476599842805
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OFFSET
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0,4
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COMMENTS
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a(n+1) gives partial sums of A033113 and second partial sums of A015518(n+1). Binomial transform of {0,0,1,1,4,4,16,16,...}.
Partial sums of floor(3^n/8) = round(3^n/8). - Mircea Merca, Dec 28 2010
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LINKS
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FORMULA
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G.f.: x^2/((1-x)^2*(1-3*x)*(1+x)).
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) + 3*a(n-4).
a(n) = Sum_{k=0..n} floor((n-k)/2)*3^k = Sum_{k=0..n} floor(k/2)*3^(n-k).
a(n) = round((3*3^n - 4*n - 4)/16) = floor((3*3^n - 4*n - 3)/16) = ceiling((3*3^n - 4*n - 5)/16) = round((3*3^n - 4*n - 3)/16).
a(n) = a(n-2) + (3^(n-1)-1)/2, n > 2. (End)
a(n) = (floor(3^(n+1)/8) - floor((n+1)/2))/2. - Seiichi Manyama, Dec 22 2023
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MAPLE
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A097137 := proc(n) add( floor(3^i/8), i=0..n) ; end proc:
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MATHEMATICA
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CoefficientList[Series[x^2/((1-x)^2(1-3x)(1+x)), {x, 0, 30}], x] (* Harvey P. Dale, Mar 11 2011 *)
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PROG
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(PARI) my(x='x+O('x^30)); concat([0, 0], Vec(x^2/((1-x)^2*(1-3*x)*(1+x)))) \\ G. C. Greubel, Jul 14 2019
(Sage) (x^2/((1-x)^2*(1-3*x)*(1+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 14 2019
(GAP) a:=[0, 0, 1, 4];; for n in [5..30] do a[n]:=4*a[n-1]-2*a[n-2]-4*a[n-3] +3*a[n-4]; od; a; # G. C. Greubel, Jul 14 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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