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A175776 Partial sums of floor(n^2/15). 1
0, 0, 0, 0, 1, 2, 4, 7, 11, 16, 22, 30, 39, 50, 63, 78, 95, 114, 135, 159, 185, 214, 246, 281, 319, 360, 405, 453, 505, 561, 621, 685, 753, 825, 902, 983, 1069, 1160, 1256, 1357, 1463, 1575, 1692, 1815, 1944, 2079, 2220, 2367, 2520, 2680, 2846 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

There are several sequences of integers of the form floor(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,9,11,12,15,16,24).

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

FORMULA

a(n) = round((2*n+1)*(n^2 + n - 14)/90).

a(n) = floor((n-2)*(2*n^2 + 7*n - 13)/90).

a(n) = ceiling((n+3)*(2*n^2 - 3*n - 18)/90).

a(n) = a(n-15) + (n+1)*(n-15) + 78, n > 14.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-15) - 3*a(n-16) + 3*a(n-17) - a(n-18); a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=2, a(6)=4, a(7)=7, a(8)=11, a(9)=16, a(10)=22, a(11)=30, a(12)=39, a(13)=50, a(14)=63, a(15)=78, a(16)=95, a(17)=114. - Harvey P. Dale, May 04 2012

G.f.: x^4*(x+1)*(x^2 - x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x-1)^4*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)). [Colin Barker, Oct 27 2012]

EXAMPLE

a(15) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 + 15 = 78.

MAPLE

seq(round((2*n+1)*(n^2+n-14)/90), n=0..50)

MATHEMATICA

Accumulate[Floor[Range[0, 60]^2/15]] (* or *) LinearRecurrence[{3, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 3, -1}, {0, 0, 0, 0, 1, 2, 4, 7, 11, 16, 22, 30, 39, 50, 63, 78, 95, 114}, 60] (* Harvey P. Dale, May 04 2012 *)

PROG

(MAGMA) [Round((2*n+1)*(n^2+n-14)/90): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

CROSSREFS

Sequence in context: A212366 A309838 A334251 * A177176 A005689 A212365

Adjacent sequences:  A175773 A175774 A175775 * A175777 A175778 A175779

KEYWORD

nonn,easy

AUTHOR

Mircea Merca, Dec 04 2010

STATUS

approved

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Last modified October 1 12:59 EDT 2020. Contains 337443 sequences. (Running on oeis4.)