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 A154260 #52 Mar 11 2025 12:29:40
%S A154260 0,7,9,30,34,69,75,124,132,195,205,282,294,385,399,504,520,639,657,
%T A154260 790,810,957,979,1140,1164,1339,1365,1554,1582,1785,1815,2032,2064,
%U A154260 2295,2329,2574,2610,2869,2907,3180,3220,3507,3549,3850,3894,4209,4255,4584
%N A154260 Numbers of the form m*(4*m +- 1)/2.
%C A154260 Also integers of the form Sum_{k = 1..j} k/4 = j*(j + 1)/8. - _Alonso del Arte_, Jan 20 2012
%C A154260 Numbers h such that 32*h + 1 is a square. - _Bruno Berselli_, Mar 30 2014
%C A154260 The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(16*n))*(1 - q^(16*n-7))*(1 - q^(16*n-9)) = 1 - q^7 - q^9 + q^30 + q^34 - q^69 - q^75 + + - - .... - _Peter Bala_, Dec 24 2024
%H A154260 G. C. Greubel, <a href="/A154260/b154260.txt">Table of n, a(n) for n = 1..1000</a>
%H A154260 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A154260 From _R. J. Mathar_, Jan 07 2009: (Start)
%F A154260 A139274 UNION A139275.
%F A154260 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A154260 G.f.: x^2*(7 + 2x + 7x^2)/((1+x)^2*(1-x)^3). (End)
%F A154260 From _G. C. Greubel_, Sep 08 2016: (Start)
%F A154260 a(n) = (1/4)*(8*n^2 + 6*(-1)^n*n - 8*n - 3*(-1)^n + 3).
%F A154260 E.g.f.: (1/4)*( (3 + 8*x^2)*exp(x) - 3*(1 + 2*x)*exp(-x) ). (End)
%F A154260 From _Amiram Eldar_, Mar 17 2022: (Start)
%F A154260 Sum_{n>=2} 1/a(n) = 8 - (sqrt(2)+1)*Pi.
%F A154260 Sum_{n>=2} (-1)^n/a(n) = 2*sqrt(2)*log(sqrt(2)+1) - 8*(1-log(2)). (End)
%F A154260 a(n) = (n-1)*(4*n-3)/2 if n is odd and a(n) = n*(4*n-1)/2 if n is even. - _Chai Wah Wu_, Mar 11 2025
%t A154260 Select[Union[Flatten[Table[{n (4n - 1)/2, n (4n + 1)/2}, {n, 0, 199}]]], IntegerQ] (* _Alonso del Arte_, Jan 20 2012 *)
%o A154260 (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 A154260 (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 A154260 (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
%o A154260 (Python)
%o A154260 def A154260(n): return (n>>1)*((n<<2)+(-3 if n&1 else -1)) # _Chai Wah Wu_, Mar 11 2025
%Y A154260 Cf. A000217, A001318, A074378, A057569, A057570.
%Y A154260 Cf. similar sequences listed in A219257 and A299645.
%K A154260 nonn,easy
%O A154260 1,2
%A A154260 _Vladimir Joseph Stephan Orlovsky_, Jan 06 2009