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A178710 Partial sums of floor(4^n/7).

%I A178710 #35 Sep 08 2022 08:45:54
%S A178710 0,2,11,47,193,778,3118,12480,49929,199725,798911,3195656,12782636,
%T A178710 51130558,204522247,818089003,3272356029,13089424134,52357696554,
%U A178710 209430786236,837723144965,3350892579881,13403570319547,53614281278212,214457125112872
%N A178710 Partial sums of floor(4^n/7).
%C A178710 Partial sums of A037521.
%H A178710 Vincenzo Librandi, <a href="/A178710/b178710.txt">Table of n, a(n) for n = 1..700</a>
%H A178710 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 A178710 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4,1,-5,4).
%F A178710 a(n) = round((8*4^n - 14*n - 13)/42).
%F A178710 a(n) = floor((8*4^n - 14*n - 8)/42).
%F A178710 a(n) = ceiling((8*4^n - 14*n - 18)/42).
%F A178710 a(n) = round((8*4^n - 14*n - 8)/42).
%F A178710 a(n) = a(n-3) + 3*4^(n-2) - 1, n > 3.
%F A178710 a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3) - 5*a(n-4) + 4*a(n-5), n > 5.
%F A178710 G.f.: x^2*(2+x)/ ( (1-4*x)*(1+x+x^2)*(1-x)^2 ).
%e A178710 a(4) = 0 + 2 + 9 + 36 = 47.
%p A178710 A178710 := proc(n) add( floor(4^i/7),i=0..n) ; end proc:
%t A178710 Accumulate[Floor[4^Range[30]/7]] (* or *) LinearRecurrence[{5,-4,1,-5,4},{0,2,11,47,193},30] (* _Harvey P. Dale_, Aug 15 2015 *)
%o A178710 (Magma) [Round((8*4^n-14*n-13)/42): n in [1..30]]; // _Vincenzo Librandi_, Jun 21 2011
%o A178710 (PARI) vector(30, n, ((8*4^n-14*n-8)/42)\1) \\ _G. C. Greubel_, Jan 25 2019
%o A178710 (Sage) [floor((8*4^n-14*n-8)/42) for n in (1..30)] # _G. C. Greubel_, Jan 25 2019
%Y A178710 Cf. A037521.
%K A178710 nonn,less
%O A178710 1,2
%A A178710 _Mircea Merca_, Dec 26 2010