A004766 Numbers whose binary expansion ends 01.

5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

These are the numbers for which zeta(2*x+1) needs just 3 terms to be evaluated. - Jorge Coveiro, Dec 16 2004

The binary representation of a(n) has exactly the same number of 0's and 1's as the binary representation of a(n+1). - Gil Broussard, Dec 18 2008

Number of monomials in n-th power of x^4+x^3+x^2+x+1. - Artur Jasinski, Oct 06 2008

Links

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

Crossrefs

Essentially same as A016813.

Programs

Maple

seq( 4*x+1, x=1..100 );

Mathematica

a = {}; k = x^4 + x^3 + x^2 + x + 1; m = k; Do[AppendTo[a, Length[m]]; m = Expand[m*k], {n, 1, 100}]; a (* Artur Jasinski, Oct 06 2008 *)
Select[Range[2,250],Take[IntegerDigits[#,2],-2]=={0,1}&] (* or *) LinearRecurrence[{2,-1},{5,9},70] (* Harvey P. Dale, Aug 07 2023 *)

PARI

a(n)=4*n+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 8*n-a(n-1)-2 (with a(1)=5). - Vincenzo Librandi, Nov 18 2010

From Colin Barker, Jun 24 2013: (Start)

a(n) = 2*a(n-1)-a(n-2).

G.f.: -x*(x-5) / (x-1)^2. (End)

E.g.f.: exp(x)*(1 + 4*x) - 1. - Stefano Spezia, Feb 02 2023