A097138 Convolution of 4^n and floor(n/2).

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

Author

Paul Barry, Jul 29 2004

Keywords

• easy
• nonn

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

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (5,-3,-5,4).

Crossrefs

Column k=4 of A368296.

Cf. A015521, A033114.

Programs

Magma

[(4^(n+2)-30*n+9*(-1)^n-25)/180: n in [0..30]]; // Vincenzo Librandi, May 31 2011

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 *)

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