A365907 Smallest nonnegative integer that is not the sum of fewer than n signed Lucas numbers.

0, 1, 5, 16, 63, 262, 1105, 4676, 19803, 83882, 355325, 1505176, 6376023, 27009262, 114413065, 484661516, 2053059123, 8696898002, 36840651125, 156059502496, 661078661103, 2800374146902, 11862575248705, 50250675141716, 212865275815563, 901711778403962

Offset: 0

Author

Mike Speciner, Sep 22 2023

Keywords

• nonn
• easy

Comments

Signed Lucas numbers are the union of A000032 and A061084.

Examples

			a(0) = 0, the sum of 0 Lucas numbers.
a(1) = 1 = A000032(1), the sum of 1 Lucas number.
a(2) = 5 = 1+4 = A000032(1)+A000032(3), the sum of 2 Lucas numbers. (2, 3, and 4 need only one term, since they are Lucas numbers.)
a(4) = 63 = 1+4+11+47.
For comparison, 45 is the first sum requiring 4 positive Lucas numbers (45 = 1+4+11+29, see A004146), but here 45 = 47+2-4 requires only 3 signed Lucas numbers so that a(4) != 45.
		

Links

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

Crossrefs

Cf. A000032, A061084, A004146 (analogous with only positive Lucas numbers).

Programs

Mathematica

LinearRecurrence[{5, -3, -1}, {0, 1, 5, 16, 63}, 26] (* Amiram Eldar, Sep 26 2023 *)

Python

from sympy import lucas
a = lambda n: n if n<2 else (lucas(3*n-2)+3)//2

Formula

a(n) = n, for n<2.

a(n) = (A000032(3*n-2)+3)/2 = 1+4+Sum_{i=2..n-1} A000032(3*n-1), for n>1.

G.f.: x*(1 - 6*x^2 - x^3)/((1 - x)*(1 - 4*x - x^2)). - Stefano Spezia, Sep 25 2023