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

%I A268291 #26 Jun 02 2025 13:25:14
%S A268291 0,1,3,6,10,15,21,28,36,45,55,66,78,78,79,81,84,88,93,99,106,114,123,
%T A268291 133,144,156,156,157,159,162,166,171,177,184,192,201,211,222,234,234,
%U A268291 235,237,240,244,249,255,262,270,279,289,300,312,312,313,315,318,322,327,333,340,348
%N A268291 a(n) = Sum_{k = 0..n} (k mod 13).
%C A268291 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)).
%C A268291 Sum_{k = 0..n} (k mod m) = m*(m - 1)/2 + Sum_{k = 1..(m - 1)} k*floor((n - k)/m), m>0.
%H A268291 Shawn A. Broyles, <a href="/A268291/b268291.txt">Table of n, a(n) for n = 0..1000</a>
%H A268291 Ilya Gutkovskiy, <a href="/A268291/a268291.pdf">Extended example</a>
%H A268291 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,1,-1).
%F A268291 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)).
%F A268291 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.
%F A268291 a(n) = 6*n + r*(r-11)/2 where r = (n mod 13). - _Hoang Xuan Thanh_, Jun 02 2025
%e A268291 (see Extended example in Links section)
%e A268291 a(0)  = 0;
%e A268291 a(1)  = 0+1 = 1;
%e A268291 a(2)  = 0+1+2 = 3;
%e A268291 a(3)  = 0+1+2+3 = 6;
%e A268291 a(4)  = 0+1+2+3+4 = 10;
%e A268291 a(5)  = 0+1+2+3+4+5 = 15;
%e A268291 ...
%e A268291 a(11) = 0+1+2+3+4+5+6+7+8+9+10+11 = 66;
%e A268291 a(12) = 0+1+2+3+4+5+6+7+8+9+10+11+12 = 78;
%e A268291 a(13) = 0+1+2+3+4+5+6+7+8+9+10+11+12+0 = 78;
%e A268291 a(14) = 0+1+2+3+4+5+6+7+8+9+10+11+12+0+1 = 79;
%e A268291 a(15) = 0+1+2+3+4+5+6+7+8+9+10+11+12+0+1+2 = 81, etc.
%t A268291 Table[Sum[Mod[k, 13], {k, 0, n}], {n, 0, 60}]
%t A268291 Table[Sum[k - 13 Floor[k/13], {k, 0, n}], {n, 0, 60}]
%t A268291 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]
%t A268291 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 *)
%t A268291 Accumulate[Mod[Range[0,60],13]] (* _Harvey P. Dale_, May 10 2021 *)
%o A268291 (PARI) a(n) = sum(k = 0, n, k % 13); \\ _Michel Marcus_, Jan 31 2016
%Y A268291 Cf. A004526, A130481-A130490.
%K A268291 nonn,easy
%O A268291 0,3
%A A268291 _Ilya Gutkovskiy_, Jan 31 2016