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%I #48 Sep 17 2023 15:50:52
%S 0,3,4,13,15,30,33,54,58,85,90,123,129,168,175,220,228,279,288,345,
%T 355,418,429,498,510,585,598,679,693,780,795,888,904,1003,1020,1125,
%U 1143,1254,1273,1390,1410,1533,1554,1683,1705,1840,1863,2004
%N Numbers of the form n*(7n+-1)/2.
%C Also integers of the form sum(k = 1..n, k/7). - _Alonso del Arte_, Jan 20 2012
%C Sequence provides all integers m such that 56*m + 1 is a square. [_Bruno Berselli_, Oct 07 2015]
%H Vincenzo Librandi, <a href="/A057570/b057570.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F G.f. -x^2*(3+x+3*x^2) / ( (1+x)^2*(x-1)^3 ). - _R. J. Mathar_, Jan 25 2011
%F a(n) = +1*a(n-1)+2*a(n-2)-2*a(n-3)-1*a(n-4)+1*a(n-5). - _Joerg Arndt_, Jan 25 2011
%F a(n) = (14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16. - _Bruno Berselli_, Jan 25 2011
%F a(n)-a(n-2) = A047341(n-1) for n>2. - _Bruno Berselli_, Jan 25 2011
%F Sum_{n>=2} 1/a(n) = 14 - 2*cot(Pi/7)*Pi. - _Amiram Eldar_, Mar 17 2022
%t lst={}; s=0; Do[s+=n/7; If[Floor[s]==s, AppendTo[lst, s]], {n, 0, 7!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 06 2009 *)
%t Select[Table[Plus@@Range[n]/7, {n, 0, 199}], IntegerQ] (* _Alonso del Arte_, Jan 20 2012 *)
%t CoefficientList[Series[-x (3 + x + 3 x^2) / ((1 + x)^2 (x - 1)^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)
%t LinearRecurrence[{1,2,-2,-1,1},{0,3,4,13,15},50] (* _Harvey P. Dale_, Sep 17 2023 *)
%o (PARI) a(n)=(14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A074378, A001318, A057569, A154260, A154292, A154293, A047341.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Oct 04 2000
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