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

1,3

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).

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A057357.

Sequence in context: A075349 A156024 A234363 * A194201 A194203 A261222

Adjacent sequences: A194226 A194227 A194228 * A194230 A194231 A194232

Keyword

nonn,easy

Author

Clark Kimberling, Aug 19 2011

Status

approved