A178872 Partial sums of round(4^n/7).

0, 1, 3, 12, 49, 195, 780, 3121, 12483, 49932, 199729, 798915, 3195660, 12782641, 51130563, 204522252, 818089009, 3272356035, 13089424140, 52357696561, 209430786243, 837723144972, 3350892579889, 13403570319555, 53614281278220, 214457125112881

Offset

0,3

Comments

a(n) (prefixed with a 0) and its higher order differences define the following infinite array:

0, 0, 1, 3, 12, 49,..

0, 1, 2, 9, 37, 146,...

1, 1, 7, 28, 109, 439... - Paul Curtz, Jun 08 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (3,3,4).

Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

Formula

a(n) = round((8*4^n+1)/42) = round((4*4^n-4)/21).

a(n) = floor((4*4^n+5)/21).

a(n) = ceiling((4*4^n-4)/21).

a(n) = a(n-3) + 3*4^(n-2) = a(n-3) + A164346(n-2) for n > 2.

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

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

a(n+1) - 4*a(n) = A049347(n). - Paul Curtz, Jun 08 2011

Example

a(3)=0+1+2+9=12.

Maple

A178872 := proc(n) add( round(4^i/7), i=0..n) ; end proc:

Mathematica

Join[{a = b = 0}, Table[c = 4^n - a - b; a = b; b = c, {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
Accumulate[Round[4^Range[0, 30]/7]] (* or *) LinearRecurrence[{3, 3, 4}, {0, 1, 3}, 30] (* Harvey P. Dale, Feb 18 2023 *)

Prog

(Magma) [Floor((4*4^n+5)/21): n in [0..30]]; // Vincenzo Librandi, May 01 2011
(PARI) a(n) = (4^(n+1)+5)\21; \\ Altug Alkan, Oct 05 2017

Crossrefs

Cf. A049347, A164346.

Sequence in context: A060113 A134589 A371436 * A323263 A344136 A248347

Adjacent sequences: A178869 A178870 A178871 * A178873 A178874 A178875

Keyword

nonn,less,easy

Author

Mircea Merca, Dec 28 2010

Status

approved