A246552 - OEIS

%I #40 Dec 17 2023 07:00:38

%S 0,0,1,2,1,1,2,3,2,2,3,4,3,3,4,5,4,4,5,6,5,5,6,7,6,6,7,8,7,7,8,9,8,8,

%T 9,10,9,9,10,11,10,10,11,12,11,11,12,13,12,12,13,14,13,13,14,15,14,14,

%U 15,16,15,15,16,17,16,16,17,18,17,17,18,19,18,18,19,20,19,19,20,21,20,20,21,22,21,21,22,23,22,22,23

%N 2-adic valuation of the number of involutions of n (A000085).

%H Vincenzo Librandi, <a href="/A246552/b246552.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F a(n) = a(n-1) + a(n-4) - a(n-5).

%F G.f.: x^2*(1+x-x^2)/((1-x)^2*(1+x)*(1+x^2)).

%F a(n) = (3 - (-1)^n - (1+3*i)*(-i)^n - (1-i*3)*i^n + 2*n)/8 where i=sqrt(-1). - _Colin Barker_, Oct 16 2015

%F a(n) = (2*n+3-2*cos(n*Pi/2)-cos(n*Pi)-6*sin(n*Pi/2))/8. - _Wesley Ivan Hurt_, Oct 01 2017

%F a(n) = n - 2*floor(n/4) - floor((n+3)/4). - _Ridouane Oudra_, Dec 11 2023

%p seq(n-2*floor(n/4)-floor((n+3)/4), n=0..100) ; # _Ridouane Oudra_, Dec 11 2023

%t CoefficientList[Series[x^2 (1 + x - x^2)/((1 - x)^2 (1 + x) (1 + x^2)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Sep 06 2014 *)

%t LinearRecurrence[{1,0,0,1,-1},{0,0,1,2,1},100] (* _Harvey P. Dale_, Jun 13 2016 *)

%o (PARI) N=166; x='x+O('x^N);

%o v=Vec(serlaplace(exp(x+x^2/2)));

%o vector(#v,n,valuation(v[n],2))

%o (PARI) concat([0,0],Vec(x^2*(1+x-x^2)/((1-x)^2*(1+x)*(1+x^2))+O(x^166)))

%o (Magma) I:=[0, 0, 1, 2, 1]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..100]]; // _Vincenzo Librandi_, Sep 06 2014

%o (PARI) a(n) = (3 - (-1)^n - (1+3*I)*(-I)^n - (1-I*3)*I^n + 2*n)/8 \\ _Colin Barker_, Oct 16 2015

%Y Cf. A000085 (involutions).

%Y Cf. A011371 (2-adic valuation of n!), A007814 (2-adic valuation of derangements (A000166)).

%K nonn,easy

%O 0,4

%A _Joerg Arndt_, Sep 06 2014