A194229 Partial sums of A057357.

0, 1, 2, 4, 6, 9, 12, 15, 19, 23, 28, 33, 39, 45, 51, 58, 65, 73, 81, 90, 99, 108, 118, 128, 139, 150, 162, 174, 186, 199, 212, 226, 240, 255, 270, 285, 301, 317, 334, 351, 369, 387, 405, 424, 443, 463, 483, 504, 525, 546, 568, 590, 613, 636, 660, 684, 708

Offset: 1

Author

Clark Kimberling, Aug 19 2011

Keywords

• nonn
• easy

Examples

			G.f. = x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + ... - _Michael Somos_, Sep 13 2023
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Crossrefs

Cf. A057357.

Programs

Mathematica

r = 3/7;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]    (* A057357 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194229 *)
Table[Sum[Floor[3*k/7], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Nov 03 2017 *)
a[ n_] := Floor[(n^2 + n)*3/14]; (* Michael Somos, Sep 13 2023 *)

PARI

concat(0, Vec(x^2*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x+x^2+x^3+x^4 +x^5+x^6)) + O(x^100))) \\ Colin Barker, Jan 09 2016

PARI

a(n) = sum(k=1, n, 3*k\7); \\ Michel Marcus, Nov 03 2017

PARI

{a(n) = (n^2+n)*3\14}; /* Michael Somos, Sep 13 2023 */

Formula

G.f.: x^2*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x+x^2+x^3+x^4+x^5+x^6)). - Colin Barker, Jan 09 2016

G.f.: x^2*(1-x^6) / ((1-x)^2*(1-x^2)*(1-x^7)). - Michael Somos, Sep 13 2023