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A194229 Partial sums of A057357.

%I A194229 #18 Sep 14 2023 00:47:22
%S A194229 0,1,2,4,6,9,12,15,19,23,28,33,39,45,51,58,65,73,81,90,99,108,118,128,
%T A194229 139,150,162,174,186,199,212,226,240,255,270,285,301,317,334,351,369,
%U A194229 387,405,424,443,463,483,504,525,546,568,590,613,636,660,684,708
%N A194229 Partial sums of A057357.
%H A194229 Colin Barker, <a href="/A194229/b194229.txt">Table of n, a(n) for n = 1..1000</a>
%H A194229 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,1,-2,1).
%F A194229 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
%F A194229 G.f.: x^2*(1-x^6) / ((1-x)^2*(1-x^2)*(1-x^7)). - _Michael Somos_, Sep 13 2023
%e A194229 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
%t A194229 r = 3/7;
%t A194229 a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
%t A194229 Table[a[n], {n, 1, 90}]    (* A057357 *)
%t A194229 s[n_] := Sum[a[k], {k, 1, n}]
%t A194229 Table[s[n], {n, 1, 100}]   (* A194229 *)
%t A194229 Table[Sum[Floor[3*k/7], {k, 1, n}], {n, 1, 50}] (* _G. C. Greubel_, Nov 03 2017 *)
%t A194229 a[ n_] := Floor[(n^2 + n)*3/14]; (* _Michael Somos_, Sep 13 2023 *)
%o A194229 (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
%o A194229 (PARI) a(n) = sum(k=1, n, 3*k\7); \\ _Michel Marcus_, Nov 03 2017
%o A194229 (PARI) {a(n) = (n^2+n)*3\14}; /* _Michael Somos_, Sep 13 2023 */
%Y A194229 Cf. A057357.
%K A194229 nonn,easy
%O A194229 1,3
%A A194229 _Clark Kimberling_, Aug 19 2011