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A335741
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Number of Pell numbers (A000129) <= n.
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2
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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
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OFFSET
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0,2
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COMMENTS
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The sequence is constant on the interval A000129(k) < n <= A000129(k+1).
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LINKS
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FORMULA
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a(n) = 1+floor(log_alpha(2*sqrt(2)*n+1)), n>=0, where alpha=1+sqrt(2).
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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