A175864 Partial sums of ceiling(n^2/19).

0, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 33, 41, 50, 61, 73, 87, 103, 121, 140, 162, 186, 212, 240, 271, 304, 340, 379, 421, 466, 514, 565, 619, 677, 738, 803, 872, 945, 1021, 1102, 1187, 1276, 1369, 1467, 1569, 1676, 1788, 1905, 2027, 2154, 2286

Offset

0,3

Comments

There are several sequences of integers of the form ceiling(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,8,10,11,12, 14,15,16,19,20,23,24).

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,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).

Formula

a(n) = round((2*n+1)*(n^2 + n + 30)/114).

a(n) = floor((2*n^3 + 3*n^2 + 61*n + 78)/114).

a(n) = ceiling((2*n^3 + 3*n^2 + 61*n - 18)/114).

a(n) = round((2*n^3 + 3*n^2 + 61*n)/114).

a(n) = a(n-19) + (n+1)*(n-19) + 140, n > 18.

G.f.: x*(x+1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^6 - x^3 + 1)/((x-1)^4*(x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)). - Colin Barker, Oct 26 2012

Example

a(19) = 0 + 1 + 1 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 11 + 12 + 14 + 16 + 18 + 19 = 140.

Maple

seq(round((2*n^3+3*n^2+61*n)/114), n=0..50)

Mathematica

Accumulate[Ceiling[Range[0, 50]^2/19]] (* Harvey P. Dale, Mar 19 2018 *)

Prog

(Magma) [Round((2*n+1)*(n^2+n+30)/114): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011
(PARI) a(n)=(2*n^3+3*n^2+61*n+78)\114 \\ Charles R Greathouse IV, Jul 06 2017

Crossrefs

Sequence in context: A039855 A175868 A035950 * A133153 A100673 A224216

Adjacent sequences: A175861 A175862 A175863 * A175865 A175866 A175867

Keyword

nonn,easy

Author

Mircea Merca, Dec 05 2010

Status

approved