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A175869 Partial sums of ceiling(n^2/23). 1
0, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 44, 53, 63, 75, 88, 103, 119, 137, 157, 179, 202, 228, 256, 286, 318, 353, 390, 430, 472, 517, 565, 616, 670, 727, 787, 850, 917, 987, 1061, 1138, 1219, 1304, 1393, 1485, 1582, 1683, 1788, 1897 (list; graph; refs; listen; history; text; internal format)
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..900

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

FORMULA

a(n) = round((2*n+1)*(n^2 + n + 42)/138).

a(n) = floor((2*n^3 + 3*n^2 + 85*n + 90)/138).

a(n) = ceiling((2*n^3 + 3*n^2 + 85*n - 6)/138).

a(n) = a(n-23) + (n+1)*(n-23) + 202.

G.f.: x*(x + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^6 - x^3 + 1)*(x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x - 1)^4*(x^22 + x^21 + x^20 + x^19 + 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(23) = 0 + 1 + 1 + 1 +  1 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 12 + 13 + 15 + 16 + 18 + 20 + 22 + 23 = 202.

MAPLE

seq(round((2*n+1)*(n^2+n+42)/138), n=0..50)

MATHEMATICA

Accumulate[Ceiling[Range[0, 50]^2/23]] (* Harvey P. Dale, Feb 04 2019 *)

PROG

(MAGMA) [Floor((2*n^3+3*n^2+85*n+90)/138): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

(PARI) a(n)=(2*n^3+3*n^2+85*n+90)\138 \\ Charles R Greathouse IV, Jul 06 2017

CROSSREFS

Sequence in context: A134953 A175870 A114829 * A007279 A034891 A143611

Adjacent sequences:  A175866 A175867 A175868 * A175870 A175871 A175872

KEYWORD

nonn,easy

AUTHOR

Mircea Merca, Dec 05 2010

STATUS

approved

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Last modified March 1 03:09 EST 2021. Contains 341732 sequences. (Running on oeis4.)