A097137 Convolution of 3^n and floor(n/2).

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

Offset: 0

Author

Paul Barry, Jul 29 2004

Keywords

• nonn

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). - Mircea Merca, Dec 28 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Crossrefs

Column k=3 of A368296.

Cf. A033113.

Programs

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

Magma

[Round((3*3^n-4*n-4)/16): n in [0..30]]; // Vincenzo Librandi, Jun 25 2011

Maple

A097137 := proc(n) add( floor(3^i/8),i=0..n) ; end proc:

Mathematica

CoefficientList[Series[x^2/((1-x)^2(1-3x)(1+x)),{x,0,30}],x]  (* Harvey P. Dale, Mar 11 2011 *)

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

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).

From Mircea Merca, Dec 26 2010: (Start)

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