A177332 Partial sums of round(n^2/29).

0, 0, 0, 0, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 35, 43, 52, 62, 73, 85, 99, 114, 131, 149, 169, 191, 214, 239, 266, 295, 326, 359, 394, 432, 472, 514, 559, 606, 656, 708, 763, 821, 882, 946, 1013, 1083, 1156, 1232, 1311, 1394, 1480

Offset: 0

Author

Mircea Merca, Dec 10 2010

Keywords

• nonn
• easy

Comments

The round function is defined here by round(x) = floor(x + 1/2).

There are several sequences of integers of the form round(n^2/k) for whose partial sums we can establish identities as following (only for k = 2, ..., 9, 11, 12, 13, 16, 17, 19, 20, 28, 29, 36, 44).

Examples

			a(17) = 0 + 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 62.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..905
• 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,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).

Crossrefs

Cf. A177100, A177116.

Programs

Magma

[Floor((n+4)*(2*n^2-5*n+21)/174): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Maple

seq(round(n*(n+1)*(2*n+1)/174),n=0..50)

Mathematica

Accumulate[Table[Round[n^2/29],{n,0,60}]] (* Harvey P. Dale, Dec 18 2010 *)

PARI

a(n)=(2*n^3+3*n^2+n+84)\174 \\ Charles R Greathouse IV, Apr 06 2012

Python

def A177332(n): return (n*(n*(2*n + 3) + 1) + 84)//174 # Chai Wah Wu, Jan 31 2023

Formula

a(n) = round(n*(n+1)*(2*n+1)/174).

a(n) = floor((n+4)*(2*n^2 - 5*n + 21)/174).

a(n) = ceiling((n-3)*(2*n^2 + 9*n + 28)/174).

a(n) = a(n-29) + (n+1)*(n-29) + 266, n > 28.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-29) - 3*a(n-30) + 3*a(n-31) - a(n-32). - R. J. Mathar, Dec 13 2010

G.f.: x^4*(x+1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^10 - x^6 + x^5 - x^4 + 1)/((x-1)^4*(x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + 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, Apr 06 2012