A335741 Number of Pell numbers (A000129) <= n.

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 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, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Author

Ovidiu Bagdasar, Jun 20 2020

Keywords

• nonn

Comments

The sequence is constant on the interval A000129(k) < n <= A000129(k+1).

Examples

			The Pell numbers A000129 are 0,1,2,5,12,29,70,...
We have a(2)=a(3)=a(4)=3, since there are three Pell numbers less than or equal to 2,3 and 4, respectively.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

Crossrefs

Cf. A000129 (Pell Numbers), A108852 (Fibonacci), A130245 (Lucas), A130253 (Jacobsthal).

Partial sums of A105349.

Programs

Mathematica

Block[{a = 2, b = -1, nn = 105, u, v = {}}, u = {0, 1}; 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 11 2021 *)
Module[{pn=LinearRecurrence[{2,1},{0,1},9],nn=100},Accumulate[Table[If[ MemberQ[ pn,n],1,0],{n,0,nn}]]] (* Harvey P. Dale, Apr 10 2022 *)

Formula

a(n) = 1+floor(log_alpha(2*sqrt(2)*n+1)), n>=0, where alpha=1+sqrt(2).