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A097137
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Convolution of 3^n and floor(n/2).
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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). [From Mircea Merca, Dec 28 2010]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (4,-2,-4,3).
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FORMULA
| 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) = ceil((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. [From Mircea Merca, Dec 26 2010]
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MAPLE
| A097137 := proc(n) add( floor(3^i/8), i=0..n) ; end proc:
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MATHEMATICA
| CoefficientList[Series[x^2/((1-x)^2(1-3x)(1+x)), {x, 0, 45}], x] (* From Harvey P. Dale, Mar 11 2011 *)
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PROG
| (MAGMA) [Round((3*3^n-4*n-4)/16): n in [0..30]]; // Vincenzo Librandi, Jun 25 2011
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CROSSREFS
| Cf. A033113.
Sequence in context: A097894 A065835 A198643 * A083377 A047115 A125068
Adjacent sequences: A097134 A097135 A097136 * A097138 A097139 A097140
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KEYWORD
| nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Jul 29 2004
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