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

Author

Casey Mongoven, May 21 2007

Keywords

• nonn
• base

Examples

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

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.

Crossrefs

Cf. A000032, A014417, A116543, A130311.

Programs

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 *)

Extensions

Definition corrected by Casey Mongoven, May 29 2010

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