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A268291 a(n) = Sum_{k = 0..n} (k mod 13). 1
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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

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Last modified May 26 17:55 EDT 2018. Contains 304622 sequences. (Running on oeis4.)