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

0,3

Links

Table of n, a(n) for n=0..26.

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A001076 (with Fibonacci numbers).

Sequence in context: A084615 A181331 A268400 * A339232 A196489 A049674

Adjacent sequences: A364751 A364752 A364753 * A364755 A364756 A364757

Keyword

nonn,easy

Author

Mike Speciner, Oct 20 2023

Status

approved