|
%I #52 Dec 22 2023 10:35:55
%S 0,0,1,7,44,266,1599,9597,57586,345520,2073125,12438755,74632536,
%T 447795222,2686771339,16120628041,96723768254,580342609532,
%U 3482055657201,20892333943215,125354003659300,752124021955810,4512744131734871,27076464790409237,162458788742455434,974752732454732616
%N Partial sums of (1/5)*floor(6^n/7).
%C Partial sums of A033116.
%H Vincenzo Librandi, <a href="/A178719/b178719.txt">Table of n, a(n) for n = 0..500</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_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-5,-7,6).
%F a(n) = (1/5)*round((24*6^n - 70*n - 49)/140).
%F a(n) = (1/5)*floor((12*6^n - 35*n - 12)/70).
%F a(n) = (1/5)*ceiling((12*6^n - 35*n - 37)/70).
%F a(n) = (1/5)*round((12*6^n - 35*n - 12)/70).
%F a(n) = a(n-2) + (6^(n-1) - 1)/5, n > 1.
%F a(n) = 7*a(n-1) - 5*a(n-2) - 7*a(n-3) + 6*a(n-4), n > 3.
%F G.f.: x^2 / ( (1+x)*(1-6*x)*(1-x)^2 ).
%F a(n) = (24*6^n - 70*n + 25*(-1)^n - 49)/700. - _Bruno Berselli_, Feb 18 2011
%F a(n) = (floor(6^(n+1)/35) - floor((n+1)/2))/5. - _Seiichi Manyama_, Dec 22 2023
%e a(3) = (1/5)*(floor(6^1/7) + floor(6^2/7) + floor(6^3/7) = (1/5)*(0+5+30) = (1/5)*35 = 7.
%p A178719 := proc(n) add( floor(6^i/7)/5,i=0..n) ; end proc:
%t f[n_] := Floor[6^n/7]/5; Accumulate@ Array[f, 22]
%t CoefficientList[Series[x^2/((1+x)(1-6x)(1-x)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2014 *)
%o (Magma) [(1/5)*Floor((12*6^n-35*n-12)/70): n in [0..30]]; // _Vincenzo Librandi_, Jun 21 2011
%o (PARI) vector(30, n, n--; (((12*6^n-35*n-12)/70)\1)/5) \\ _G. C. Greubel_, Jan 24 2019
%o (Sage) [floor((12*6^n-35*n-12)/70)/5 for n in (0..30)] # _G. C. Greubel_, Jan 24 2019
%Y Column k=6 of A368296.
%Y Cf. A033116.
%K nonn,less
%O 0,4
%A _Mircea Merca_, Dec 26 2010
| 