A376022 - OEIS

%I #21 Nov 29 2024 05:12:25

%S 1,-1,-3,-13,7,2,0,-1,-3,-6,-15,59,9,4,2,0,-1,-3,-5,-8,-14,-40,53,15,

%T 8,5,3,1,-1,-3,-5,-7,-10,-16,-31,-225,43,19,11,7,4,2,0,-1,-3,-5,-7,

%U -10,-14,-21,-37,-130,88,32,19,13,9,6,4,2,0,-1,-3,-5,-7,-9,-12,-16,-22

%N a(1) = 1, for n >= 2, a(n) = -1 + floor((n*a(n - 1)) / (n + a(n - 1))).

%C For x =< -2 and for some k, a(3*k*(k - 1)/2 + x) = -(2*x + 4).

%H John Tyler Rascoe, <a href="/A376022/b376022.txt">Table of n, a(n) for n = 1..10000</a>

%F For x >= 0, k >= 2*x + 3 - floor((sqrt(9 + 8*x) - 1) / 2), a(3*k*(k-1)/2 + x) = -(2*x + 3).

%F For k >= 2, a(3*k*(k - 1)/2 - 1) = -1.

%e a(1) = 1.

%e a(2) = -1 + floor(2*a(1) / (2 + a(1))) = -1 + floor(2/3) = -1.

%e a(3) = -1 + floor(-3/2) = -3.

%e a(4) = -1 + floor(-12/1) = -13.

%e a(5) = -1 + floor(-65/-8) = 7.

%e and so on.

%t a[1] = 1; a[n_] := a[n] = -1 + Floor[n*a[n-1]/(n + a[n-1])]; Array[a, 100] (* _Amiram Eldar_, Sep 06 2024 *)

%o (Python)

%o from itertools import count, islice

%o def a_gen():

%o     a = 1

%o     for n in count(2):

%o         yield a

%o         b = -1+(n*a)//(n+a)

%o         a = b

%o A376022_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Sep 17 2024

%o (PARI) lista(nn)= my(a=-2); vector(nn, n, a=-1+floor(n*a/(n+a))); \\ _Ruud H.G. van Tol_, Nov 28 2024

%Y Cf. A000217, A003056, A005408, A005448, A045943, A124011, A139482, A143208, A290768.

%K sign,easy

%O 1,3

%A _Ctibor O. Zizka_, Sep 06 2024