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%I #48 Dec 22 2023 10:35:51
%S 0,1,8,58,408,2859,20016,140116,980816,6865717,48060024,336420174,
%T 2354941224,16484588575,115392120032,807744840232,5654213881632,
%U 39579497171433,277056480200040,1939395361400290,13575767529802040,95030372708614291,665212608960300048,4656488262722100348,32595417839054702448
%N Partial sums of floor(7^n/8)/6.
%C Partial sums of A033117.
%H Vincenzo Librandi, <a href="/A178730/b178730.txt">Table of n, a(n) for n = 1..1000</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 (8,-6,-8,7).
%F 6*a(n) = round((7*7^n - 24*n - 16)/48).
%F 6*a(n) = floor((7*7^n - 24*n - 7)/48).
%F 6*a(n) = ceiling((7*7^n - 24*n - 25)/48).
%F 6*a(n) = round((7*7^n - 24*n - 7)/48).
%F a(n) = a(n-2) + (7^(n-1) - 1)/6, n > 2.
%F a(n) = 8*a(n-1) - 6*a(n-2) - 8*a(n-3) + 7*a(n-4), n > 4.
%F G.f.: x^2/((1+x)*(1-7*x)*(1-x)^2).
%F a(n) = (7^(n+1) - 24*n + 9*(-1)^n - 16)/288. - _Bruno Berselli_, Jan 11 2011
%F a(n) = (floor(7^(n+1)/48) - floor((n+1)/2))/6. - _Seiichi Manyama_, Dec 22 2023
%e a(3) = (1/6)*(floor(7/8) + floor(7^2/8) + floor(7^3/8)) = (1/6)*(0+6+42) = 8.
%p A178730 := proc(n) add( floor(7^i/8)/6,i=0..n) ; end proc:
%t CoefficientList[Series[x/((1+x)(1-7x)(1-x)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2014 *)
%o (Magma) [Floor((7*7^n-24*n-7)/48)/6: n in [1..30]]; // _Vincenzo Librandi_, Jun 21 2011
%o (PARI) vector(30, n, (((7^(n+1)-24*n-7)/48)\1)/6) \\ _G. C. Greubel_, Jan 24 2019
%o (Sage) [floor((7^(n+1)-24*n-7)/48)/6 for n in (1..30)] # _G. C. Greubel_, Jan 24 2019
%Y Column k=7 of A368296.
%Y Cf. A033117.
%K nonn,less
%O 1,3
%A _Mircea Merca_, Dec 26 2010
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