A130310 Minimal (or "greedy") Lucas representation of n, in which L(0) = 2 and L(2) = 3 are not allowed in the same representation (hence the correct representation of the integer 5 is 1010 rather than 101). A binary system of integers with Lucas numbers (A000032) as a base.

0, 10, 1, 100, 1000, 1010, 1001, 10000, 10010, 10001, 10100, 100000, 100010, 100001, 100100, 101000, 101010, 101001, 1000000, 1000010, 1000001, 1000100, 1001000, 1001010, 1001001, 1010000, 1010010, 1010001, 1010100, 10000000, 10000010, 10000001, 10000100, 10001000

Offset

0,2

References

Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers, Singapore, World Scientific, 1997, pp. 73-77.

Edouard Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège, Vol. 41 (1972), pp. 179-182.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Ron Knott, Using Powers of Phi to represent Integers.

Example

a(9) = 10001 because 7 + 2 = 9.
a(10) is 10100 because 7 + 3 = 10.

Mathematica

a[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; FromDigits @ IntegerDigits[Total[2^s], 2]]; Array[a, 30, 0] (* Amiram Eldar, Feb 17 2022 *)

Crossrefs

Cf. A000032, A014417, A116543, A130311.

Sequence in context: A038303 A178870 A075505 * A333685 A288050 A288503

Adjacent sequences: A130307 A130308 A130309 * A130311 A130312 A130313

Keyword

nonn,base

Author

Casey Mongoven, May 21 2007

Extensions

Definition corrected by Casey Mongoven, May 29 2010

a(0) and more terms from Amiram Eldar, Feb 17 2022

Status

approved