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A177205 Partial sums of round(n^2/17). 1
0, 0, 0, 1, 2, 3, 5, 8, 12, 17, 23, 30, 38, 48, 60, 73, 88, 105, 124, 145, 169, 195, 223, 254, 288, 325, 365, 408, 454, 503, 556, 613, 673, 737, 805, 877, 953, 1034, 1119, 1208, 1302, 1401, 1505, 1614, 1728, 1847, 1971, 2101, 2237, 2378, 2525 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..890

Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

FORMULA

a(n)=round(n*(n+1)*(2*n+1)/102).

a(n)=floor((2*n^3+3*n^2+n+36)/102).

a(n)=ceil((2*n^3+3*n^2+n-36)/102).

a(n)=a(n-17)+(n+1)*(n-17)+105 , n>16.

a(n)= +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-17) -3*a(n-18) +3*a(n-19) -a(n-20) with g.f. x^3 *(1+x) *(x^12-2*x^11+2*x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+2*x^2-2*x+1) / ( (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) *(x-1)^4 ). - R. J. Mathar, Dec 13 2010

EXAMPLE

a(17) = 0+0+0+1+1+1+2+3+4+5+6+7+8+10+12+13+15+17 = 105.

MAPLE

seq(round(n*(n+1)*(2*n+1)/102), n=0..50)

PROG

(MAGMA) [Floor((2*n^3+3*n^2+n+36)/102): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

CROSSREFS

Cf. A177100, A177116.

Sequence in context: A104664 A022856 A089071 * A275580 A175829 A241552

Adjacent sequences:  A177202 A177203 A177204 * A177206 A177207 A177208

KEYWORD

nonn,easy

AUTHOR

Mircea Merca, Dec 10 2010

STATUS

approved

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Last modified May 30 08:46 EDT 2017. Contains 287302 sequences.