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

1,5

Comments

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Hugo Pfoertner)

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Formula

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

a(n) = A102302(n) for n >= 7. - conjectured by Hugo Pfoertner, Oct 10 2018; proved by Max Alekseyev, Mar 16 2026

a(n) = A116921(n) for n>=3. - conjectured by R. J. Mathar, Aug 29 2025; proved by Max Alekseyev, Mar 16 2026

Conjectured by Colin Barker, Oct 10 2018; proved by Max Alekseyev, Mar 16 2026: (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)

Maple

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) ; # R. J. Mathar, Mar 20 2010

Mathematica

Log2[Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 100]]]] (* Paolo Xausa, Feb 01 2024, after Nick Hobson in A173300 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 0, 1, 1, 2, 1, 3}, 100] (* Paolo Xausa, Mar 22 2026 *)

Prog

(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

Crossrefs

Cf. A102302, A173300.

Sequence in context: A163281 A307857 A116921 * A093068 A097357 A342016

Adjacent sequences: A173986 A173987 A173988 * A173990 A173991 A173992

Keyword

nonn,easy

Author

J. Lowell, Mar 04 2010

Extensions

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

Status

approved