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 A172131 Partial sums of floor(n^2/9) (A056838). 1
 0, 0, 0, 1, 2, 4, 8, 13, 20, 29, 40, 53, 69, 87, 108, 133, 161, 193, 229, 269, 313, 362, 415, 473, 537, 606, 681, 762, 849, 942, 1042, 1148, 1261, 1382, 1510, 1646, 1790, 1942, 2102, 2271, 2448, 2634, 2830, 3035, 3250, 3475, 3710, 3955, 4211, 4477, 4754 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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. Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,1,-3,3,-1). FORMULA a(n) = Sum_{k=0..n} floor(k^2/9). a(n) = round((2*n^3 + 3*n^2 - 15*n - 9)/54). a(n) = round((2*n^3 + 3*n^2 - 15*n - 8)/54). a(n) = floor((2*n^3 + 3*n^2 - 15*n + 18)/54). a(n) = ceiling((2*n^3 + 3*n^2 - 15*n - 34)/54). a(n) = a(n-9) + (n-4)^2 + 4, n > 8. G.f.: x^3*(x+1)*(x^2 - x + 1)^2/((x-1)^4*(x^2 + x + 1)*(x^6 + x^3 + 1)). [Colin Barker, Oct 26 2012] EXAMPLE a(6) = 8 = 0 + 0 + 0 + 1 + 1 + 2 + 4. MAPLE a:= n-> round((2*n^3+3*n^2-15*n-9)/54): seq (a(n), n=0..50); MATHEMATICA Accumulate[Floor[Range[0, 50]^2/9]] (* or *) LinearRecurrence[{3, -3, 1, 0, 0, 0, 0, 0, 1, -3, 3, -1}, {0, 0, 0, 1, 2, 4, 8, 13, 20, 29, 40, 53}, 60] (* Harvey P. Dale, Jan 10 2020 *) PROG (MAGMA) [Round((2*n^3+3*n^2-15*n-9)/54): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011 CROSSREFS Cf. A056838. Sequence in context: A084684 A011907 A056133 * A173721 A164482 A247587 Adjacent sequences:  A172128 A172129 A172130 * A172132 A172133 A172134 KEYWORD nonn,easy AUTHOR Mircea Merca, Nov 19 2010 STATUS approved

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Last modified November 30 03:03 EST 2021. Contains 349416 sequences. (Running on oeis4.)