A178744 Partial sums of floor(4^n/9).

0, 0, 1, 8, 36, 149, 604, 2424, 9705, 38832, 155340, 621373, 2485508, 9942048, 39768209, 159072856, 636291444, 2545165797, 10180663212, 40722652872, 162890611513, 651562446080, 2606249784348, 10424999137421, 41699996549716, 166799986198896

Offset

0,4

Comments

Partial sums of A037597.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..170

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 (5,-4,1,-5,4).

Formula

a(n) = round((8*4^n - 24*n - 17)/54).

a(n) = floor(4*(4^n - 3*n - 1)/27).

a(n) = ceiling((4*4^n - 12*n - 13)/27).

a(n) = round(4*(4^n - 3*n - 1)/27).

a(n) = a(n-3) + (7*4^(n-2) - 4)/3 , n > 3.

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

G.f.: x^2*(1+3*x) / ( (1-4*x)*(1+x+x^2)*(1-x)^2 ).

Example

a(4) = 0 + 1 + 7 + 28 = 36.

Maple

A178744 := proc(n) add( floor(4^i/9), i=0..n) ; end proc:

Mathematica

Table[Floor[4*(4^n-3*n-1)/27], {n, 0, 30}] (* G. C. Greubel, Jan 24 2019 *)

Prog

(Magma) [&+[Floor(4^k/9): k in [0..n]]: n in [0..25]]; // Bruno Berselli, Apr 26 2011
(PARI) vector(30, n, n--; (4*(4^n-3*n-1)/27)\1) \\ G. C. Greubel, Jan 24 2019
(Sage) [floor(4*(4^n-3*n-1)/27) for n in (0..30)] # G. C. Greubel, Jan 24 2019

Crossrefs

Cf. A037597.

Sequence in context: A000427 A000428 A083597 * A200707 A344207 A055918

Adjacent sequences: A178741 A178742 A178743 * A178745 A178746 A178747

Keyword

nonn

Author

Mircea Merca, Dec 26 2010

Status

approved