login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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