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
The sum of 3 consecutive terms is a(n) + a(n+1) + a(n+2) = 4^(n+1). - Klaus Purath, Dec 22 2025
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
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 (3,3,4).
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
KEYWORD
nonn,less,easy
AUTHOR
Mircea Merca, Dec 28 2010
STATUS
approved