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A173721 Partial sums of A056833. 1
0, 0, 1, 2, 4, 8, 13, 20, 29, 41, 55, 72, 93, 117, 145, 177, 214, 255, 301, 353, 410, 473, 542, 618, 700, 789, 886, 990, 1102, 1222, 1351, 1488, 1634, 1790, 1955, 2130, 2315, 2511, 2717, 2934, 3163, 3403, 3655, 3919, 4196, 4485, 4787, 5103, 5432, 5775, 6132 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,1,-3,3,-1).

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

FORMULA

a(n) = Sum_{k=0..n} round(k^2/7).

a(n) = round((2*n^3 + 3*n^2 + n)/42).

a(n) = a(n-7) + (n-3)^2 + 4, n > 6.

From R. J. Mathar, Nov 26 2010: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-7) - 3*a(n-8) + 3*a(n-9) - a(n-10).

G.f.: x^2*(1+x)*(x^2-x+1)^2 / ( (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x-1)^4 ).

a(n) = n*(n+1)*(2*n+1)/42 - b(n)/7 where b(n) = 0, 1, -2, 0, 2, -1, 0, ... with period 7. (End)

EXAMPLE

a(5) = round(1/7) + round(4/7) + round(9/7) + round(16/7) + round(25/7) = 0+1+1+2+4 = 8.

MAPLE

A173721 := proc(n) a := n*(n+1)*(2*n+1)/42 ; b := op( 1+(n mod 7), [0, 1, -2, 0, 2, -1, 0]) ; a-b/7 ; end proc:

seq(A173721(n), n=0..80) ; # R. J. Mathar, Nov 26 2010

MATHEMATICA

Table[Round[(2*n^3 + 3*n^2 + n)/42], {n, 0, 50}] (* G. C. Greubel, Nov 29 2016 *)

PROG

(MAGMA) a056833:=func< n | Round(n^2/7) >; [ &+[ a056833(j): j in [0..n] ]: n in [0..60] ]; // Klaus Brockhaus, Nov 26 2010

(PARI) a(n)=(2*n^3+3*n^2+n)\/42 \\ Charles R Greathouse IV, Nov 29 2016

CROSSREFS

Cf. A056833 (nearest integer to n^2/7).

Sequence in context: A011907 A056133 A172131 * A164482 A247587 A061866

Adjacent sequences:  A173718 A173719 A173720 * A173722 A173723 A173724

KEYWORD

nonn,easy

AUTHOR

Mircea Merca, Nov 26 2010

STATUS

approved

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Last modified May 6 21:35 EDT 2021. Contains 343597 sequences. (Running on oeis4.)