A363070 Take the terms 0..n of the infinite Fibonacci word A003849, regard them as a number in Fibonacci base.

0, 1, 2, 3, 6, 10, 17, 28, 45, 74, 120, 194, 315, 510, 826, 1337, 2163, 3501, 5665, 9167, 14833, 24000, 38834, 62835, 101669, 164505, 266175, 430681, 696857, 1127538, 1824396, 2951935, 4776331, 7728267, 12504599, 20232867, 32737467, 52970334, 85707802, 138678137, 224385940, 363064078

Offset: 0

Author

Gandhar Joshi, May 16 2023

Keywords

• base
• nonn

Examples

			0 ->                            0 -> a(0) = 0,
0,1 ->                         01 -> a(1) = 1,
0,1,0 ->                      010 -> a(2) = 2,
0,1,0,0 ->                   0100 -> a(3) = 3,
0,1,0,0,1 ->                01001 -> a(4) = 6,
0,1,0,0,1,0 ->             010010 -> a(5) = 10,
0,1,0,0,1,0,1 ->          0100101 -> a(6) = 17,
0,1,0,0,1,0,1,0 ->       01001010 -> a(7) = 28,
0,1,0,0,1,0,1,0,0 ->    010010100 -> a(8) = 45,
0,1,0,0,1,0,1,0,0,1 -> 0100101001 -> a(9) = 74.
		

Links

• Gandhar Joshi, Table of n, a(n) for n = 0..1000
• Agnibha Banerjee, Python Program, May 16 2023.

Crossrefs

Cf. A003849 (Fibonacci word), A005713, A189920 (Zeckendorf digits).

Cf. A182028.

Programs

Python

# see linked program

Python

def aupto(n): # produces n terms, indices 0..n-1
    F1, F, a = [0], [0, 1], [0, 1]
    while len(F) < n:
        F1, F = F, F+F1
    [a.append(a[-2]+a[-1]+F[i]+F[i-1]) for i in range(2, n)]
    return a
print(aupto(42)) # Michael S. Branicky, May 17 2023

Formula

a(n) = Sum_{i=0..n} A003849(i)*Fibonacci(n-i+2).

a(n) = a(n-1) + a(n-2) + A003849(n) + A003849(n-1).

a(n) = a(n-1) + a(n-2) + A005713(n-1). - Kevin Ryde, May 20 2023