A156002 - OEIS

%I #45 May 06 2024 08:30:03

%S 0,1,6,44,311,2178,15250,106755,747288,5231022,36617161,256320132,

%T 1794240932,12559686533,87917805738,615424640176,4307972481243,

%U 30155807368710,211090651580982,1477634561066887,10343441927468220

%N Partial sums of round(7^n/9).

%H Vincenzo Librandi, <a href="/A156002/b156002.txt">Table of n, a(n) for n = 0..100</a>

%H 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.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (8,-7,1,-8,7).

%F a(n) = round((7*7^n - 6*n + 2)/54) = round((7*7^n - 6*n - 7)/54).

%F a(n) = floor((7*7^n - 6*n + 11)/54).

%F a(n) = ceiling((7*7^n - 6*n - 7)/54).

%F a(n) = a(n-3) + (19*7^(n-2) - 1)/3, n > 2.

%F a(n) = 8*a(n-1) - 7*a(n-2) + a(n-3) - 8*a(n-4) + 7*a(n-5), n > 4.

%F G.f.: -x*(1 - 2*x + 3*x^2)/((7*x-1)*(1+x+x^2)*(x-1)^2).

%e a(3) = 0 + 1 + 5 + 38 = 44.

%p A156002 := proc(n) add( round(7^i/9),i=0..n) ; end proc:

%t CoefficientList[Series[-x*(1 - 2*x + 3*x^2)/((7*x - 1)*(1 + x + x^2)*(x - 1)^2),{x, 0, 40}], x] (* or *)

%t LinearRecurrence[{8, -7, 1, -8, 7}, {0, 1, 6, 44, 311}, 40] (* _Stefano Spezia_, Sep 02 2018 *)

%t Accumulate[Round[7^Range[0,20]/9]] (* _Harvey P. Dale_, Apr 24 2021 *)

%o (Magma) [Floor((7*7^n-6*n+11)/54): n in [0..40]]; // _Vincenzo Librandi_, Apr 27 2011

%Y Cf. A178826.

%K nonn,easy,less

%O 0,3

%A _Mircea Merca_, Dec 28 2010