|
%I #42 Mar 17 2022 07:04:06
%S 0,7,9,30,34,69,75,124,132,195,205,282,294,385,399,504,520,639,657,
%T 790,810,957,979,1140,1164,1339,1365,1554,1582,1785,1815,2032,2064,
%U 2295,2329,2574,2610,2869,2907,3180,3220,3507,3549,3850,3894,4209,4255,4584
%N Numbers of the form m*(4*m +- 1)/2.
%C Also integers of the form Sum_{k = 1..j} k/4 = j*(j + 1)/8. - _Alonso del Arte_, Jan 20 2012
%C Numbers h such that 32*h + 1 is a square. - _Bruno Berselli_, Mar 30 2014
%H G. C. Greubel, <a href="/A154260/b154260.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 From _R. J. Mathar_, Jan 07 2009: (Start)
%F A139274 UNION A139275.
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F G.f.: x^2*(7 + 2x + 7x^2)/((1+x)^2*(1-x)^3). (End)
%F From _G. C. Greubel_, Sep 08 2016: (Start)
%F a(n) = (1/4)*(8*n^2 + 6*(-1)^n*n - 8*n - 3*(-1)^n + 3).
%F E.g.f.: (1/4)*( (3 + 8*x^2)*exp(x) - 3*(1 + 2*x)*exp(-x) ). (End)
%F From _Amiram Eldar_, Mar 17 2022: (Start)
%F Sum_{n>=2} 1/a(n) = 8 - (sqrt(2)+1)*Pi.
%F Sum_{n>=2} (-1)^n/a(n) = 2*sqrt(2)*log(sqrt(2)+1) - 8*(1-log(2)). (End)
%t Select[Union[Flatten[Table[{n (4n - 1)/2, n (4n + 1)/2}, {n, 0, 199}]]], IntegerQ] (* _Alonso del Arte_, Jan 20 2012 *)
%o (PARI) print1(0);forstep(n=2,1e2,2,print1(", "n*(4*n-1)/2", "n*(4*n+1)/2)) \\ _Charles R Greathouse IV_, Jan 20 2012
%o (PARI) print1(s=0);for(n=1,1e3,s+=n/4;if(denominator(s)==1,print1(s", "))) \\ _Charles R Greathouse IV_, Jan 20 2012
%o (Magma) k:=8; f:=func<n | n*(k*n+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..25]]; // _Bruno Berselli_, Nov 14 2012
%Y Cf. A000217, A001318, A074378, A057569, A057570.
%Y Cf. similar sequences listed in A299645.
%K nonn,easy
%O 1,2
%A _Vladimir Joseph Stephan Orlovsky_, Jan 06 2009
|
