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A342089
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Numbers that have two representations as the sum of distinct non-consecutive Lucas numbers (A000032).
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1
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5, 12, 16, 23, 30, 34, 41, 45, 52, 59, 63, 70, 77, 81, 88, 92, 99, 106, 110, 117, 121, 128, 135, 139, 146, 153, 157, 164, 168, 175, 182, 186, 193, 200, 204, 211, 215, 222, 229, 233, 240, 244, 251, 258, 262, 269, 276, 280, 287, 291, 298, 305, 309, 316, 320, 327
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OFFSET
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1,1
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COMMENTS
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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.
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).
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
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LINKS
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EXAMPLE
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5 is a term since is has two representations: L(0) + L(2) = 2 + 3 and L(1) + L(3) = 1 + 4.
12 is a term since is has two representations: L(1) + L(5) = 1 + 11 and L(0) + L(2) + L(4) = 2 + 3 + 7.
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MAPLE
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L:= [seq(combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1), n=0..40)]:
f1:= proc(n, m) option remember;
if n = 0 then return 1 fi;
if m <= 0 then 0
elif L[m] <= n then procname(n - L[m], m-2) + procname(n, m-1)
else procname(n, m-1)
fi
end proc:
filter:= n -> f1(n, ListTools:-BinaryPlace(L, n+1))=2:
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MATHEMATICA
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L = Table[Fibonacci[n+1] + Fibonacci[n-1], {n, 0, 40}];
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]]];
filterQ[n_] := f1[n, FirstPosition[L, b_ /; b > n+1][[1]]-1] == 2;
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PROG
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(Java) See David C. Luo's GitHub link.
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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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