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A097138
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Convolution of 4^n and floor(n/2).
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1
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0, 0, 1, 5, 22, 90, 363, 1455, 5824, 23300, 93205, 372825, 1491306, 5965230, 23860927, 95443715, 381774868, 1527099480, 6108397929, 24433591725, 97734366910, 390937467650, 1563749870611, 6254999482455, 25019997929832
(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 A033114 and second partial sums of A015521.
Partial sums of 1/3*floor(4^n/5). [From Mircea Merca (mircea(AT)teacher.com), Dec 26 2010]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index to sequences with linear recurrences with constant coefficients, signature (5,-3,-5,4)
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FORMULA
| G.f.: x^2/((1-x)*(1-4*x)*(1-x^2)).
a(n) = sum{k=0..n, floor((n-k)/2)4^k} = sum{k=0..n, floor(k/2)*4^(n-k)}.
a(n) = 5*a(n-1)-3*a(n-2)-5*a(n-3)+4*a(n-4).
3*a(n) = round((16*4^n-30*n-25)/60) = floor((8*4^n-15*n-8)/30) = ceil((8*4^n-15*n-17)/30) = round((8*4^n-15*n-8)/30). a(n) = a(n-2)+(4^(n-1)-1)/3 , n>1 . [From Mircea Merca (mircea(AT)teacher.com), Dec 26 2010]
a(n) = (4^(n+2)-30*n+9*(-1)^n-25)/180. - Bruno Berselli, Dec 27 2010
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EXAMPLE
| a(3) = 1/3*floor(4^0/5)+1/3*floor(4^1/5)+1/3*floor(4^2/5) +1/3*floor(4^3/5) = 0 + 0 + 1 + 4 = 5
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MAPLE
| A097138 := proc(n) (4^(n+2)-30*n+9*(-1)^n-25)/180 ; end proc: # R. J. Mathar, Jan 08 2011
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PROG
| (MAGMA) [(4^(n+2)-30*n+9*(-1)^n-25)/180: n in [0..30]]; // Vincenzo Librandi, May 31 2011
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CROSSREFS
| Sequence in context: A108072 A081892 A128566 * A050185 A085812 A172061
Adjacent sequences: A097135 A097136 A097137 * A097139 A097140 A097141
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Jul 29 2004
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