cp's OEIS Frontend

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

%I A178872 #40 Feb 18 2023 10:01:20
%S A178872 0,1,3,12,49,195,780,3121,12483,49932,199729,798915,3195660,12782641,
%T A178872 51130563,204522252,818089009,3272356035,13089424140,52357696561,
%U A178872 209430786243,837723144972,3350892579889,13403570319555,53614281278220,214457125112881
%N A178872 Partial sums of round(4^n/7).
%C A178872 a(n) (prefixed with a 0) and its higher order differences define the following infinite array:
%C A178872 0, 0, 1,  3,  12,  49,..
%C A178872 0, 1, 2,  9,  37, 146,...
%C A178872 1, 1, 7, 28, 109, 439... - _Paul Curtz_, Jun 08 2011
%H A178872 Vincenzo Librandi, <a href="/A178872/b178872.txt">Table of n, a(n) for n = 0..500</a>
%H A178872 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,4).
%H A178872 Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a> J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
%F A178872 a(n) = round((8*4^n+1)/42) = round((4*4^n-4)/21).
%F A178872 a(n) = floor((4*4^n+5)/21).
%F A178872 a(n) = ceiling((4*4^n-4)/21).
%F A178872 a(n) = a(n-3) + 3*4^(n-2) = a(n-3) + A164346(n-2) for n > 2.
%F A178872 a(n) = 3*a(n-1) + 3*a(n-2) + 4*a(n-3) for n > 2.
%F A178872 G.f.: -x/((4*x-1)*(x^2+x+1)).
%F A178872 a(n+1) - 4*a(n) = A049347(n). - _Paul Curtz_, Jun 08 2011
%e A178872 a(3)=0+1+2+9=12.
%p A178872 A178872 := proc(n) add( round(4^i/7),i=0..n) ; end proc:
%t A178872 Join[{a = b = 0}, Table[c = 4^n - a - b; a = b; b = c, {n, 0, 100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 28 2011 *)
%t A178872 Accumulate[Round[4^Range[0,30]/7]] (* or *) LinearRecurrence[{3,3,4},{0,1,3},30] (* _Harvey P. Dale_, Feb 18 2023 *)
%o A178872 (Magma) [Floor((4*4^n+5)/21): n in [0..30]]; // _Vincenzo Librandi_, May 01 2011
%o A178872 (PARI) a(n) = (4^(n+1)+5)\21; \\ _Altug Alkan_, Oct 05 2017
%Y A178872 Cf. A049347, A164346.
%K A178872 nonn,less,easy
%O A178872 0,3
%A A178872 _Mircea Merca_, Dec 28 2010