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A177237 Partial sums of round(n^2/19). 1
0, 0, 0, 0, 1, 2, 4, 7, 10, 14, 19, 25, 33, 42, 52, 64, 77, 92, 109, 128, 149, 172, 197, 225, 255, 288, 324, 362, 403, 447, 494, 545, 599, 656, 717, 781, 849, 921, 997, 1077, 1161, 1249, 1342, 1439, 1541, 1648, 1759, 1875, 1996, 2122, 2254 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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..895

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-2)*(n+3)*(2*n+1)/114).

a(n) = floor((2*n^3 + 3*n^2 - 11*n + 42)/114).

a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 54)/114).

a(n) = round((2*n^3 + 3*n^2 - 11*n)/114).

a(n) = a(n-19) + (n+1)*(n-19) + 128, n > 18.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-19) - 3*a(n-20) + 3*a(n-21) - a(n-22) with g.f. x^4*(1+x)*(x^4 - x^3 + x^2 - x + 1)*(x^8 - x^7 + x^6 - x^4 + x^2 - x + 1) / ( (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) *(x-1)^4 ). - R. J. Mathar, Dec 13 2010

EXAMPLE

a(19) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 3 + 3 + 4 + 5 + 6 + 8 + 9 + 10 + 12 + 13 + 15 + 17 + 19 = 128.

MAPLE

seq(round((2*n^3+3*n^2-11*n)/114), n=0..50)

PROG

(MAGMA) [Floor((2*n^3+3*n^2-11*n+42)/114): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

CROSSREFS

Cf. A177100, A177116.

Sequence in context: A023536 A196126 A024536 * A094281 A076101 A288243

Adjacent sequences:  A177234 A177235 A177236 * A177238 A177239 A177240

KEYWORD

nonn,easy

AUTHOR

Mircea Merca, Dec 10 2010

STATUS

approved

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Last modified May 31 00:29 EDT 2020. Contains 334747 sequences. (Running on oeis4.)