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Partial sums of floor(4^n/9).
2

%I #34 Jul 13 2024 11:43:23

%S 0,0,1,8,36,149,604,2424,9705,38832,155340,621373,2485508,9942048,

%T 39768209,159072856,636291444,2545165797,10180663212,40722652872,

%U 162890611513,651562446080,2606249784348,10424999137421,41699996549716,166799986198896

%N Partial sums of floor(4^n/9).

%C Partial sums of A037597.

%H Vincenzo Librandi, <a href="/A178744/b178744.txt">Table of n, a(n) for n = 0..170</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 (5,-4,1,-5,4).

%F a(n) = round((8*4^n - 24*n - 17)/54).

%F a(n) = floor(4*(4^n - 3*n - 1)/27).

%F a(n) = ceiling((4*4^n - 12*n - 13)/27).

%F a(n) = round(4*(4^n - 3*n - 1)/27).

%F a(n) = a(n-3) + (7*4^(n-2) - 4)/3 , n > 3.

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

%F G.f.: x^2*(1+3*x) / ( (1-4*x)*(1+x+x^2)*(1-x)^2 ).

%e a(4) = 0 + 1 + 7 + 28 = 36.

%p A178744 := proc(n) add( floor(4^i/9),i=0..n) ; end proc:

%t Table[Floor[4*(4^n-3*n-1)/27], {n,0,30}] (* _G. C. Greubel_, Jan 24 2019 *)

%t Accumulate[Floor[4^Range[0,30]/9]] (* or *) LinearRecurrence[{5,-4,1,-5,4},{0,0,1,8,36},30] (* _Harvey P. Dale_, Jul 13 2024 *)

%o (Magma) [&+[Floor(4^k/9): k in [0..n]]: n in [0..25]]; // _Bruno Berselli_, Apr 26 2011

%o (PARI) vector(30, n, n--; (4*(4^n-3*n-1)/27)\1) \\ _G. C. Greubel_, Jan 24 2019

%o (Sage) [floor(4*(4^n-3*n-1)/27) for n in (0..30)] # _G. C. Greubel_, Jan 24 2019

%Y Cf. A037597.

%K nonn

%O 0,4

%A _Mircea Merca_, Dec 26 2010