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A342089 Numbers that have two representations as the sum of distinct non-consecutive Lucas numbers (A000032).

%I A342089 #31 Jan 05 2025 19:51:41
%S A342089 5,12,16,23,30,34,41,45,52,59,63,70,77,81,88,92,99,106,110,117,121,
%T A342089 128,135,139,146,153,157,164,168,175,182,186,193,200,204,211,215,222,
%U A342089 229,233,240,244,251,258,262,269,276,280,287,291,298,305,309,316,320,327
%N A342089 Numbers that have two representations as the sum of distinct non-consecutive Lucas numbers (A000032).
%C A342089 Brown (1969) proved that every positive number has a unique representation as a sum of non-consecutive Lucas numbers, if L(0) = 2 and L(2) = 3 do not appear simultaneously in the representation.
%C A342089 Chu et al. (2020) proved that if L(0) and L(2) are allowed to appear simultaneously, then each positive number can have at most two representations. The terms with two representations are listed in this sequence. They found that the number of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 17, 171, 1708, 17082, 170820, ..., and proved that the asymptotic density of this sequence is 1/(3*phi+1) = 0.1708203932... (A176015 - 1), where phi is the golden ratio (A001622).
%C A342089 A number n appears in the sequence if and only if the coefficient of phi^{-1} in the base-phi expansion of n is 1.  Alternatively, the last bit of the n-th term of A341722 is 1. - _Jeffrey Shallit_, May 03 2023
%H A342089 Robert Israel, <a href="/A342089/b342089.txt">Table of n, a(n) for n = 1..10000</a>
%H A342089 John L. Brown, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-3/brown.pdf">Unique representation of integers as sums of distinct Lucas numbers</a>, The Fibonacci Quarterly, Vol. 7, No. 3 (1969), pp. 243-252.
%H A342089 Hung V. Chu, David C. Luo and Steven J. Miller, <a href="https://arxiv.org/abs/2004.08316">On Zeckendorf Related Partitions Using the Lucas Sequence</a>, arXiv:2004.08316 [math.NT], 2020-2021.
%H A342089 David C. Luo, <a href="https://github.com/dluo6745/Zeckendorf-Partitions/blob/master/ZP.java">Java code for calculating non-consecutive partitions of natural numbers in any infinite integer sequence given by a second-order linear recurrence</a>, GitHub.
%H A342089 Jeffrey Shallit, <a href="https://arxiv.org/abs/2305.02672">Proving Properties of phi-Representations with the Walnut Theorem-Prover</a>, arXiv:2305.02672 [math.NT], 2023.
%e A342089 5 is a term since it has two representations: L(0) + L(2) = 2 + 3 and L(1) + L(3) = 1 + 4.
%e A342089 12 is a term since it has two representations: L(1) + L(5) = 1 + 11 and L(0) + L(2) + L(4) = 2 + 3 + 7.
%p A342089 L:= [seq(combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1), n=0..40)]:
%p A342089 f1:= proc(n, m) option remember;
%p A342089       if n = 0 then return 1 fi;
%p A342089       if m <= 0 then 0
%p A342089       elif L[m] <= n then procname(n - L[m],m-2) + procname(n, m-1)
%p A342089       else procname(n,m-1)
%p A342089       fi
%p A342089 end proc:
%p A342089 filter:= n -> f1(n,ListTools:-BinaryPlace(L,n+1))=2:
%p A342089 select(filter, [$1..1000]); # _Robert Israel_, Mar 10 2021
%t A342089 L = Table[Fibonacci[n+1] + Fibonacci[n-1], {n, 0, 40}];
%t A342089 f1[n_, m_] := f1[n, m] = If[n == 0, Return[1], Which[m <= 0, 0, L[[m]] <= n, f1[n-L[[m]], m-2] + f1[n, m-1], True, f1[n, m-1]]];
%t A342089 filterQ[n_] := f1[n, FirstPosition[L, b_ /; b > n+1][[1]]-1] == 2;
%t A342089 Select[Range[1000], filterQ] (* _Jean-François Alcover_, Aug 27 2022, after _Robert Israel_ *)
%o A342089 (Java) See David C. Luo's GitHub link.
%Y A342089 Cf. A000032, A001622, A130310, A116543, A176015, A341722.
%K A342089 nonn
%O A342089 1,1
%A A342089 _Amiram Eldar_, Feb 27 2021