This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A057570 #59 Nov 22 2024 08:34:08
%S A057570 0,3,4,13,15,30,33,54,58,85,90,123,129,168,175,220,228,279,288,345,
%T A057570 355,418,429,498,510,585,598,679,693,780,795,888,904,1003,1020,1125,
%U A057570 1143,1254,1273,1390,1410,1533,1554,1683,1705,1840,1863,2004
%N A057570 Numbers of the form n*(7n+-1)/2.
%C A057570 Also integers of the form Sum_{k = 1..n} k/7. - _Alonso del Arte_, Jan 20 2012
%C A057570 Sequence provides all integers m such that 56*m + 1 is a square. [_Bruno Berselli_, Oct 07 2015]
%C A057570 The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-3)) * (1 + x^(7*n-4)) = 1 + x^3 + x^4 + x^13 + x^15 + x^30 + x^33 + .... Cf. A363801. - _Peter Bala_, Nov 21 2024
%H A057570 Vincenzo Librandi, <a href="/A057570/b057570.txt">Table of n, a(n) for n = 1..1000</a>
%H A057570 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A057570 G.f.: -x^2*(3+x+3*x^2) / ( (1+x)^2*(x-1)^3 ). - _R. J. Mathar_, Jan 25 2011
%F A057570 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 A057570 a(n) = (14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16. - _Bruno Berselli_, Jan 25 2011
%F A057570 a(n)-a(n-2) = A047341(n-1) for n>2. - _Bruno Berselli_, Jan 25 2011
%F A057570 Sum_{n>=2} 1/a(n) = 14 - 2*cot(Pi/7)*Pi. - _Amiram Eldar_, Mar 17 2022
%t A057570 Select[Table[Plus@@Range[n]/7, {n, 0, 199}], IntegerQ] (* _Alonso del Arte_, Jan 20 2012 *)
%t A057570 CoefficientList[Series[-x (3 + x + 3 x^2) / ((1 + x)^2 (x - 1)^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)
%t A057570 LinearRecurrence[{1,2,-2,-1,1},{0,3,4,13,15},50] (* _Harvey P. Dale_, Sep 17 2023 *)
%o A057570 (PARI) a(n)=(14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A057570 Cf. A074378, A001318, A057569, A154260, A154292, A154293, A047341, A274830, A363801.
%K A057570 nonn,easy
%O A057570 1,2
%A A057570 _N. J. A. Sloane_, Oct 04 2000