A173989 a(n) is the 2-adic valuation of A173300(n).

0, 0, 1, 1, 2, 1, 3, 3, 4, 3, 5, 5, 6, 5, 7, 7, 8, 7, 9, 9, 10, 9, 11, 11, 12, 11, 13, 13, 14, 13, 15, 15, 16, 15, 17, 17, 18, 17, 19, 19, 20, 19, 21, 21, 22, 21, 23, 23, 24, 23, 25, 25, 26, 25, 27, 27, 28, 27, 29, 29, 30, 29, 31, 31, 32, 31, 33, 33, 34, 33, 35, 35, 36, 35, 37, 37, 38, 37

Offset: 1

Author

J. Lowell, Mar 04 2010

Keywords

• nonn
• changed

Comments

Conjecture: always follows the pattern A, A, A+1, A, where A is an odd number.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A102302, A173300.

Programs

Maple

From R. J. Mathar, Mar 20 2010: (Start)
A173300 := proc(n) local x,y ; x := (1+sqrt(3))/2 ; y := (1-sqrt(3))/2 ; denom(expand(x^n+y^n)) ; end proc:
A173989 := proc(n) log[2](A173300(n)) ; end proc: seq(A173989(n),n=3..100) ; (End)

Mathematica

Log2[Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 100]]]] (* Paolo Xausa, Feb 01 2024, after Nick Hobson in A173300 *)

PARI

\\ using Max Alekseyev's function in A173300
A173300(n) = denominator(2*polcoeff( lift( Mod((1+x)/2, x^2-3)^n ), 0))
for(k=1,74,print1(logint(A173300(k),2),", ")) \\ Hugo Pfoertner, Oct 10 2018

Formula

a(n) = log(A173300(n))/log(2).

Apparently a(n) = A102302(n) for n >= 7. - Hugo Pfoertner, Oct 10 2018

Conjectures from Colin Barker, Oct 10 2018: (Start)

G.f.: x^3*(1 + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).

a(n) = a(n-1) + a(n-4) - a(n-5) for n > 7.

(End)

Apparently a(n) = A116921(n) for n>=3. - R. J. Mathar, Aug 29 2025

Extensions

More terms from R. J. Mathar and Max Alekseyev, Mar 20 2010