|
%I #39 Sep 08 2022 08:44:56
%S 0,2,3,4,5,7,8,9,10,12,13,14,15,17,18,19,20,22,23,24,25,27,28,29,30,
%T 32,33,34,35,37,38,39,40,42,43,44,45,47,48,49,50,52,53,54,55,57,58,59,
%U 60,62,63,64,65,67,68,69,70,72,73,74,75,77,78,79,80,82,83,84,85,87,88,89
%N Numbers that are congruent to {0, 2, 3, 4} mod 5.
%C Complement of A016861. - _Reinhard Zumkeller_, Oct 23 2006
%H Melvyn B. Nathanson, <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.05.409">On the fractional parts of roots of positive real numbers</a>, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A027445(a(n)) mod 10 = 0. - _Reinhard Zumkeller_, Oct 23 2006
%F a(n) = floor((5n-2)/4). - _Gary Detlefs_, Mar 06 2010
%F a(n) = floor((15n-5)/12). - _Gary Detlefs_, Mar 07 2010
%F G.f.: x^2*(2+x+x^2+x^3)/((1+x)*(1+x^2)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011
%F From _Wesley Ivan Hurt_, May 14 2016: (Start)
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F a(n) = (10*n-7+(-1)^n+2*(-1)^((2*n+3+(-1)^n)/4))/8.
%F a(2n) = A047211(n), a(2n-1) = A047218(n).
%F a(n) = A047207(n+1) - 1.
%F a(n+2) = n + 2 + A002265(n) for n>0.
%F a(n+3)-a(n+2) = A177704(n) for n>0.
%F a(1-n) = - A001068(n). (End)
%F Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + sqrt(5)*log(phi)/10 - sqrt(5-2*sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - _Amiram Eldar_, Dec 07 2021
%p seq(floor(5*n-2)/4), n=1..72); # _Gary Detlefs_, Mar 06 2010
%p seq(floor((15*n-5)/12), n=1..72); # _Gary Detlefs_, Mar 07 2010
%t Flatten[Table[5*n + {0, 2, 3, 4}, {n, 0, 20}]] (* _T. D. Noe_, Nov 12 2013 *)
%o (PARI) a(n)=(5*n-2)\4 \\ _Charles R Greathouse IV_, Jun 11 2015
%o (Magma) [Floor((5*n-2)/4) : n in [1..100]]; // _Wesley Ivan Hurt_, May 14 2016
%Y Cf. A001068, A001622, A002265, A016861, A027445, A047207, A047211, A047218, A177704.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Reinhard Zumkeller_, Oct 23 2006
|
