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

1,1

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

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.

Example

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.

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 *)

Prog

(Java) See David C. Luo's GitHub link.

Crossrefs

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

Sequence in context: A314281 A314282 A314283 * A008467 A076718 A076671

Adjacent sequences: A342086 A342087 A342088 * A342090 A342091 A342092

Keyword

nonn

Author

Amiram Eldar, Feb 27 2021

Status

approved