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 A131941 Partial sums of ceiling(n^2/2) (A000982). 15
 0, 1, 3, 8, 16, 29, 47, 72, 104, 145, 195, 256, 328, 413, 511, 624, 752, 897, 1059, 1240, 1440, 1661, 1903, 2168, 2456, 2769, 3107, 3472, 3864, 4285, 4735, 5216, 5728, 6273, 6851, 7464, 8112, 8797, 9519, 10280, 11080, 11921, 12803, 13728, 14696, 15709 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Binomial transform of [0, 1, 1, 2, -2, 4, -8, 16, -32, ...]. Starting with offset 1 = (1, 3, 5, 7, ...) convolved with (1, 0, 3, 0, 5, ...). - Gary W. Adamson, Feb 16 2009 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1, p. 11. Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1). FORMULA For even n, a(n) = n*(2*n^2 +3*n +4)/12. For odd n, a(n) = (n+1)*(2*n^2 +n +3)/12. - Washington Bomfim, Jul 31 2008 From R. J. Mathar, Feb 24 2010: (Start) G.f.: x*(1+x^2)/((1+x)*(1-x)^4). a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End) From Mircea Merca, Oct 10 2010: (Start) a(n) = round((2*n^3 + 3*n^2 + 4*n)/12) = round((2*n+1)*(2*n^2 + 3*n + 3)/24) = floor((n+1)*(2*n^2 + n + 3)/12) = ceiling((2*n^3 + 3*n^2 + 4*n)/12). a(n) = a(n-2) + n^2 - n + 1, n > 1. (End) a(n) = (2*n*(2*n^2 + 3*n + 4) - 3*(-1)^n + 3)/24. - Bruno Berselli, Dec 07 2010 EXAMPLE a(3) = 8 = 0 + 1 + 2 + 5. MAPLE a(n):=round(1/(12)(2*n^(3)+3*n^(2)+4*n))  # Mircea Merca, Oct 10 2010 MATHEMATICA CoefficientList[Series[x (1 + x^2)/(1 + x)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *) PROG (PARI) a(n) = (n+[0, 1][n%2+1]) * (2*n^2 +[3, 1][n%2+1]*n +[4, 3][n%2+1])/12 \\ Washington Bomfim, Jul 31 2008 (MAGMA) [Ceiling((2*n^3+3*n^2+4*n)/12): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011 CROSSREFS Cf. A000982, A080930 (binomial transform without leading 0). Sequence in context: A225253 A254875 A025202 * A009858 A169947 A167616 Adjacent sequences:  A131938 A131939 A131940 * A131942 A131943 A131944 KEYWORD nonn,easy AUTHOR Gary W. Adamson, Oct 25 2007 STATUS approved

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Last modified August 10 08:35 EDT 2020. Contains 336368 sequences. (Running on oeis4.)