A364754 Smallest nonnegative integer not expressible by the addition and subtraction of fewer than n Lucas numbers.

0, 1, 5, 23, 99, 421, 1785, 7563, 32039, 135721, 574925, 2435423, 10316619, 43701901, 185124225, 784198803, 3321919439, 14071876561, 59609425685, 252509579303, 1069647742899, 4531100550901, 19194049946505, 81307300336923, 344423251294199, 1459000305513721, 6180424473349085

Offset: 0

Author

Mike Speciner, Oct 20 2023

Keywords

• nonn
• easy

Examples

			a(0) = 0, since 0 is expressible as the sum of 0 Lucas numbers.
a(1) = 1, since 1 is a Lucas number.
a(2) = 5, since 2, 3, and 4 are all Lucas numbers; while 5=1+4, the sum of 2 Lucas numbers.
a(3) = 23, since integers less than 23 are expressible with 2 or fewer Lucas numbers, while 23 = 1+4+18 requires 3 terms.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Crossrefs

Cf. A000032, A004146 (adding positive Lucas numbers), A365907 (adding any Lucas numbers).

Cf. A001076 (with Fibonacci numbers).

Programs

Mathematica

a[n_] := (LucasL[3*n - 1] - 1)/2; a[0] = 0; Array[a, 27, 0] (* Amiram Eldar, Oct 21 2023 *)

Python

from sympy import lucas
a = lambda n: n and (lucas(3*n-1)-1)//2

Formula

a(0) = 0.

a(n) = (A000032(3*n-1)-1)/2, for n > 0.

a(n) = 1 + Sum_{i=1..n-1} A000032(3*i), for n > 0.

G.f.: x*(1 + x^2)/((1 - x)*(1 - 4*x - x^2)). - Stefano Spezia, Oct 21 2023