A178742 Partial sums of floor(2^n/9).

0, 0, 0, 0, 1, 4, 11, 25, 53, 109, 222, 449, 904, 1814, 3634, 7274, 14555, 29118, 58245, 116499, 233007, 466023, 932056, 1864123, 3728258, 7456528, 14913068, 29826148, 59652309, 119304632, 238609279, 477218573, 954437161

Offset: 0

Author

Mircea Merca, Dec 26 2010

Keywords

• nonn
• less

Comments

Partial sums of A153234.

Examples

			a(6) = 0 + 0 + 0 + 0 + 1 + 3 + 7 = 11.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,1,4,-5,2).

Crossrefs

Cf. A153234.

Programs

Magma

[&+[Floor(2^k/9): k in [0..n]]: n in [0..25]];  // Bruno Berselli, Apr 26 2011

Magma

I:=[0,0,0,0,1,4]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+4*Self(n-4)-5*Self(n-5)+2*Self(n-6): n in [1..40]]; // Vincenzo Librandi, Mar 26 2014

Maple

A178742 := proc(n) add( floor(2^i/9),i=0..n) ; end proc:

Mathematica

CoefficientList[Series[x^4/((1-2x)(1+x)(1-x+x^2)(1-x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
LinearRecurrence[{4,-5,1,4,-5,2},{0,0,0,0,1,4},40] (* Harvey P. Dale, Jan 25 2015 *)

PARI

vector(30, n, n--; ((4*2^n-9*n+2)/18)\1) \\ G. C. Greubel, Jan 24 2019

Sage

[floor((4*2^n-9*n+2)/18) for n in (0..30)] # G. C. Greubel, Jan 24 2019

Formula

a(n) = round((8*2^n - 18*n - 9)/36).

a(n) = floor((4*2^n - 9*n + 2)/18).

a(n) = ceiling((4*2^n - 9*n - 11)/18).

a(n) = round((4*2^n - 9*n - 4)/18).

a(n) = a(n-6) + 7*2^(n-5) - 3, n > 5.

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 4*a(n-4) - 5*a(n-5) + 2*a(n-6).

G.f.: x^4 / ( (1-2*x)*(1+x)*(1-x+x^2)*(1-x)^2 ).