A245906 Numbers of the form 4n^2 + 1 or 4n^2 + 8n + 1.

5, 13, 17, 33, 37, 61, 65, 97, 101, 141, 145, 193, 197, 253, 257, 321, 325, 397, 401, 481, 485, 573, 577, 673, 677, 781, 785, 897, 901, 1021, 1025, 1153, 1157, 1293, 1297, 1441, 1445, 1597, 1601, 1761, 1765, 1933, 1937, 2113, 2117, 2301, 2305, 2497, 2501, 2701

Offset: 1

Author

Jamel Ghanouchi, Nov 13 2014

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Programs

Magma

[IsEven(n) select n^2+4*n+1 else (n+1)^2+1: n in [1..50]]; // Bruno Berselli, Dec 02 2014

Mathematica

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 *)

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

PARI

a(n)=if(n%2,(n+1)^2+1,n^2+4*n+1) \\ Charles R Greathouse IV, Nov 20 2014

Formula

a(n) = n^2 + O(n). - Charles R Greathouse IV, Nov 20 2014

G.f.: x*(5+8*x-6*x^2+x^4)/((1+x)^2*(1-x)^3). [Bruno Berselli, Dec 02 2014]

a(n) = (2*n*(n+5)-(2n+1)*(-1)^n+11)/2. [Bruno Berselli, Dec 02 2014]

Extensions

Extended by Charles R Greathouse IV, Nov 13 2014