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A178872 Partial sums of round(4^n/7). 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A060113 A134589 A371436 * A323263 A344136 A248347
KEYWORD
nonn,less,easy
AUTHOR
Mircea Merca, Dec 28 2010
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)