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

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

Offset: 1

Author

Amiram Eldar, Feb 27 2021

Keywords

• nonn

Comments

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

Examples

			5 is a term since it has two representations: L(0) + L(2) = 2 + 3 and L(1) + L(3) = 1 + 4.
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• John L. Brown, Jr., Unique representation of integers as sums of distinct Lucas numbers, The Fibonacci Quarterly, Vol. 7, No. 3 (1969), pp. 243-252.
• Hung V. Chu, David C. Luo and Steven J. Miller, On Zeckendorf Related Partitions Using the Lucas Sequence, arXiv:2004.08316 [math.NT], 2020-2021.
• David C. Luo, Java code for calculating non-consecutive partitions of natural numbers in any infinite integer sequence given by a second-order linear recurrence, GitHub.
• Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Crossrefs

Cf. A000032, A001622, A130310, A116543, A176015, A341722.

Programs

Java

See David C. Luo's GitHub link.

Maple

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:
select(filter, [$1..1000]); # Robert Israel, Mar 10 2021

Mathematica

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;
Select[Range[1000], filterQ] (* Jean-François Alcover, Aug 27 2022, after Robert Israel *)