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

0,3

Links

Gandhar Joshi, Table of n, a(n) for n = 0..1000

Agnibha Banerjee, Python Program, May 16 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

Example

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.

Prog

(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

Crossrefs

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

Cf. A182028.

Sequence in context: A099517 A026647 A026669 * A023614 A001610 A324015

Adjacent sequences: A363067 A363068 A363069 * A363071 A363072 A363073

Keyword

base,nonn

Author

Gandhar Joshi, May 16 2023

Status

approved