A022801 n-th Lucas number (A000204(n)) + n-th non-Lucas number (A090946(n+1)).

3, 8, 10, 15, 20, 28, 41, 60, 90, 138, 215, 339, 540, 863, 1385, 2229, 3594, 5802, 9374, 15153, 24503, 39631, 64109, 103713, 167793, 271476, 439238, 710682, 1149887, 1860535, 3010387, 4870886, 7881236, 12752084, 20633281, 33385325, 54018565

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Programs

Maple

with(combinat): lucas:= n->fibonacci(n+1)+fibonacci(n-1): n:=1: for k from 1 to 7 do for nonlucas from lucas(k)+1 to lucas(k+1)-1 do printf("%d, ",nonlucas+lucas(n)) :n:=n+1 od od: # C. Ronaldo (aga_new_ac(AT)hotmail.com)

Mathematica

Module[{nn=40,luc,nluc},luc=LucasL[Range[nn]];nluc=Complement[Range[ Last[ luc]],luc];Total/@Thread[{luc,Take[nluc,Length[luc]]}]] (* Harvey P. Dale, May 02 2019 *)

Python

from sympy import lucas
def A022801(n):
    def f(x):
        if x<=2: return n+1
        a, b, c = 1, 3, 0
        while b<=x:
            a, b = b, a+b
            c += 1
        return n+1+c
    m, k = n+1, f(n+1)
    while m != k: m, k = k, f(k)
    return m+lucas(n) # Chai Wah Wu, Sep 10 2024

Formula

a(n) = A000204(n) + A090946(n + 1). - Sean A. Irvine, May 21 2019

Extensions

Thanks to Karima MOUSSAOUI (bouyao(AT)wanadoo.fr), who noticed that there were two versions of this sequence, differing at about the 22nd term, Feb 28 2004

More terms from Emeric Deutsch, Jan 14 2005