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
Examples
(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.
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).
Programs
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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 *) -
PARI
a(n) = sum(k = 0, n, k % 13); \\ Michel Marcus, Jan 31 2016
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.
a(n) = 6*n + r*(r-11)/2 where r = (n mod 13). - Hoang Xuan Thanh, Jun 02 2025
Comments