A172131 Partial sums of floor(n^2/9) (A056838).

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

Offset: 0

Author

Mircea Merca, Nov 19 2010

Keywords

• nonn
• easy

Examples

			a(6) = 8 = 0 + 0 + 0 + 1 + 1 + 2 + 4.
		

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).

Crossrefs

Cf. A056838.

Programs

Magma

[Round((2*n^3+3*n^2-15*n-9)/54): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

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 *)

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]