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 A178420 Partial sums of floor(2^n/3). 3
 0, 1, 3, 8, 18, 39, 81, 166, 336, 677, 1359, 2724, 5454, 10915, 21837, 43682, 87372, 174753, 349515, 699040, 1398090, 2796191, 5592393, 11184798, 22369608, 44739229, 89478471, 178956956, 357913926, 715827867, 1431655749, 2863311514 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Essentially the same as A011377: 0 followed by the terms of A011377. - Joerg Arndt, Apr 22 2016 Partial sums of A000975(n-1). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1. Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2). FORMULA a(n) = A011377(n-1) for n >= 1. - Joerg Arndt, Apr 22 2016 a(n) = round((8*2^n - 6*n - 9)/12). a(n) = floor((4*2^n - 3*n - 4)/6). a(n) = ceiling((4*2^n - 3*n - 5)/6). a(n) = round((4*2^n - 3*n - 4)/6). a(n) = a(n-2) + 2^(n-1) - 1, n > 2. From Bruno Berselli, Jan 15 2011: (Start) a(n) = (8*2^n - 6*n - 9 + (-1)^n)/12. G.f.: x^2/((1+x)*(1-2*x)*(1-x)^2). (End) G.f.: Q(0)/(3*(1-x)^2), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013 a(n) = 2*a(n-1) + floor(n/2) for n > 1. - Bruno Berselli, Apr 30 2014 EXAMPLE a(5) = 0 + 1 + 2 + 5 + 10 = 18. MAPLE seq(round((4*2^n-3*n-4)/6), n=1..50) MATHEMATICA f[n_] := Floor[(4 2^n - 3 n - 4)/6]; f[Range[60]] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2011 *) CoefficientList[Series[x / ((1 + x) (1 - 2 x) (1 - x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *) LinearRecurrence[{3, -1, -3, 2}, {0, 1, 3, 8}, 40] (* or *) Accumulate[ Table[ Floor[ 2^n/3], {n, 40}]] (* Harvey P. Dale, Dec 24 2015 *) PROG (MAGMA) [Floor((4*2^n-3*n-4)/6): n in [1..30]]; // Vincenzo Librandi, Jun 23 2011 (PARI) a(n)=(4<

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Last modified August 8 23:02 EDT 2020. Contains 336300 sequences. (Running on oeis4.)