A345376 - OEIS

%I #12 Jun 23 2021 10:50:56

%S 0,0,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,

%T 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,

%U 5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6

%N Number of Companion Pell numbers m <= n.

%C Table 1 of Andrica 2021 paper (p. 24) refers to A002203 as "Pell-Lucas" numbers.

%H Michael De Vlieger, <a href="/A345376/b345376.txt">Table of n, a(n) for n = 0..10000</a>

%H Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, <a href="https://doi.org/10.2478/auom-2021-0002">On some new results for the generalised Lucas sequences</a>, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36. See Section 5.2, pp. 32-33, Table 3.

%e The Pell-Lucas numbers A002203 are 2, 2, 6, 14, 34, 82, ...

%e a(0)=a(1)=0, since there are no Pell-Lucas numbers less than or equal to 0 and 1, respectively.

%e a(2)=a(3)=a(4)=a(5)=2, since the first 2 Pell-Lucas numbers, 2 and 2, are less than or equal to 2, 3, 4, and 5, respectively.

%t Block[{a = 2, b = -1, nn = 64, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v]  (* _Michael De Vlieger_, Jun 16 2021 *)

%Y Cf. A002203, A108852 (Fibonacci), A130245 (Lucas), A335741 (Pell).

%K nonn,easy

%O 0,3

%A _Ovidiu Bagdasar_, Jun 16 2021