%I #22 Sep 08 2022 08:46:09
%S 5,13,17,33,37,61,65,97,101,141,145,193,197,253,257,321,325,397,401,
%T 481,485,573,577,673,677,781,785,897,901,1021,1025,1153,1157,1293,
%U 1297,1441,1445,1597,1601,1761,1765,1933,1937,2113,2117,2301,2305,2497,2501,2701
%N Numbers of the form 4n^2 + 1 or 4n^2 + 8n + 1.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = n^2 + O(n). - _Charles R Greathouse IV_, Nov 20 2014
%F G.f.: x*(5+8*x-6*x^2+x^4)/((1+x)^2*(1-x)^3). [_Bruno Berselli_, Dec 02 2014]
%F a(n) = (2*n*(n+5)-(2n+1)*(-1)^n+11)/2. [_Bruno Berselli_, Dec 02 2014]
%t fn[n_]:=Module[{c=4n^2+1},{c,c+8n}]; Flatten[Array[fn,30]] (* or *) LinearRecurrence[{1,2,-2,-1,1},{5,13,17,33,37},60] (* _Harvey P. Dale_, May 21 2015 *)
%o (PARI) list(lim)=Set(concat(vector(sqrtint((lim-1)\4),n,4*n^2+1), vector(sqrtint(lim\1+3)\2-1,n,4*n^2+8*n+1))) \\ _Charles R Greathouse IV_, Nov 13 2014
%o (PARI) a(n)=if(n%2,(n+1)^2+1,n^2+4*n+1) \\ _Charles R Greathouse IV_, Nov 20 2014
%o (Magma) [IsEven(n) select n^2+4*n+1 else (n+1)^2+1: n in [1..50]]; // _Bruno Berselli_, Dec 02 2014
%K nonn,easy
%O 1,1
%A _Jamel Ghanouchi_, Nov 13 2014
%E Extended by _Charles R Greathouse IV_, Nov 13 2014