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A097138
Convolution of 4^n and floor(n/2).
3
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
OFFSET
0,4
COMMENTS
a(n+1) gives partial sums of A033114 and second partial sums of A015521.
Partial sums of 1/3*floor(4^n/5). - Mircea Merca, Dec 26 2010
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).
From Mircea Merca, Dec 26 2010: (Start)
3*a(n) = round((16*4^n-30*n-25)/60) = floor((8*4^n-15*n-8)/30) = ceiling((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. (End)
a(n) = (4^(n+2)-30*n+9*(-1)^n-25)/180. - Bruno Berselli, Dec 27 2010
a(n) = (floor(4^(n+1)/15) - floor((n+1)/2))/3. - Seiichi Manyama, Dec 22 2023
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.
MAPLE
A097138 := proc(n) (4^(n+2)-30*n+9*(-1)^n-25)/180 ; end proc: # R. J. Mathar, Jan 08 2011
MATHEMATICA
LinearRecurrence[{5, -3, -5, 4}, {0, 0, 1, 5}, 30] (* Harvey P. Dale, Sep 17 2017 *)
PROG
(Magma) [(4^(n+2)-30*n+9*(-1)^n-25)/180: n in [0..30]]; // Vincenzo Librandi, May 31 2011
CROSSREFS
Column k=4 of A368296.
Sequence in context: A108072 A081892 A128566 * A105467 A208736 A050185
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 29 2004
STATUS
approved