A365907 -id:A365907 - cp's OEIS frontend

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

Mike Speciner, Oct 20 2023

			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.

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

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

Cf. A001076 (with Fibonacci numbers).

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

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

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

A367816 Number of terms in a shortest sequence of Lucas numbers that sum to n, allowing Lucas numbers with negative indices.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 3, 2, 3, 3, 3, 3, 4, 3, 2, 3, 3, 2, 1, 2, 2, 2

Offset: 0

Mike Speciner, Dec 01 2023

			For n = 0, the empty sequence sums to 0, so a(0) = 0.
For n = 1, 2, 3, 4, 7, 11, 18, each n is a Lucas number, so a(n) = 1.
The first n needing a negative-index Lucas number is 17 = 18 + -1; a(17) = 2.

Cf. A000032 Lucas numbers; A061084 negative index Lucas numbers.
A116543 is the similar sequence where negative index Lucas numbers are not allowed.
a(A365907(n)) is the first occurrence of n.

from itertools import count
def  a(n) :
  """For integer n, the least number of Lucas terms required to sum to n."""
  f = [2,1];    # Lucas numbers, starting with Lucas(0)
  while f[-1] <= (n or 1) :
    f.append(f[-2]+f[-1]);
  a = [0 for _ in range(f[-1]+1)];
  for i in f :
    a[i] = 1;
  for c in count(2) :
    if not all(a[4:]) :
      for i in range(4,f[-1]) :
        if not a[i] :
          for j in f :
            if j >= i :
              break;
            if a[i-j] == c-1 :
              a[i] = c;
              break;
          if not a[i]:
            for j in f[1::2] :
              if i+j >= len(a) :
                break;
              if a[i+j] == c-1 :
                a[i] = c;
                break;
    else :
      break;
  return a[n];

a(0) = 0; a(A000032(n)) = 1.

For n > 0, a(n) = 1+min(a(n-Lucas(k))) where k ranges over Z.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula