A173721 Partial sums of A056833.

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

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 A359850 A349216

Adjacent sequences: A173718 A173719 A173720 * A173722 A173723 A173724

Keyword

nonn,easy

Author

Mircea Merca, Nov 26 2010

Status

approved