A268291 a(n) = Sum_{k = 0..n} (k mod 13).

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 78, 79, 81, 84, 88, 93, 99, 106, 114, 123, 133, 144, 156, 156, 157, 159, 162, 166, 171, 177, 184, 192, 201, 211, 222, 234, 234, 235, 237, 240, 244, 249, 255, 262, 270, 279, 289, 300, 312, 312, 313, 315, 318, 322, 327, 333, 340, 348

Offset

0,3

Comments

More generally, the ordinary generating function for the Sum_{k = 0..n} (k mod m) is (Sum_{k = 1..(m - 1)} k*x^k)/((1 - x^m)*(1 - x)).

Sum_{k = 0..n} (k mod m) = m*(m - 1)/2 + Sum_{k = 1..(m - 1)} k*floor((n - k)/m), m>0.

Links

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

Ilya Gutkovskiy, Extended example

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

Formula

G.f.: (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + 9*x^9 + 10*x^10 + 11*x^11 + 12*x^12)/((1 - x^13)*(1 - x)).

a(n) = 12*floor((n - 12)/13) + 11*floor((n - 11)/13) + 10*floor((n - 10)/13) + 9*floor((n - 9)/13) + 8*floor((n - 8)/13) + 7*floor((n - 7)/13) + 6*floor((n - 6)/13) + 5*floor((n - 5)/13) + 4*floor((n - 4)/13) + 3*floor((n - 3)/13) + 2*floor((n - 2)/13) + floor((n - 1)/13) + 78.

Example

(see Extended example in Links section)
a(0)  = 0;
a(1)  = 0+1 = 1;
a(2)  = 0+1+2 = 3;
a(3)  = 0+1+2+3 = 6;
a(4)  = 0+1+2+3+4 = 10;
a(5)  = 0+1+2+3+4+5 = 15;
...
a(11) = 0+1+2+3+4+5+6+7+8+9+10+11 = 66;
a(12) = 0+1+2+3+4+5+6+7+8+9+10+11+12 = 78;
a(13) = 0+1+2+3+4+5+6+7+8+9+10+11+12+0 = 78;
a(14) = 0+1+2+3+4+5+6+7+8+9+10+11+12+0+1 = 79;
a(15) = 0+1+2+3+4+5+6+7+8+9+10+11+12+0+1+2 = 81, etc.

Mathematica

Table[Sum[Mod[k, 13], {k, 0, n}], {n, 0, 60}]
Table[Sum[k - 13 Floor[k/13], {k, 0, n}], {n, 0, 60}]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 78}, 61]
CoefficientList[Series[(x + 2 x^2 + 3 x^3 + 4 x^4 + 5 x^5 + 6 x^6 + 7 x^7 + 8 x^8 + 9 x^9 + 10 x^10 + 11 x^11 + 12 x^12) / ((1 - x^13) (1 - x)), {x, 0, 70}], x] (* Vincenzo Librandi, Jan 31 2016 *)
Accumulate[Mod[Range[0, 60], 13]] (* Harvey P. Dale, May 10 2021 *)

Prog

(PARI) a(n) = sum(k = 0, n, k % 13); \\ Michel Marcus, Jan 31 2016

Crossrefs

Cf. A004526, A130481-A130490.

Sequence in context: A130490 A033444 A061791 * A105336 A130910 A105337

Adjacent sequences: A268288 A268289 A268290 * A268292 A268293 A268294

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jan 31 2016

Status

approved