A377704 a(n) = binomial(Fibonacci(n)+Fibonacci(n+1)-2,Fibonacci(n)-1).

1, 1, 3, 15, 330, 50388, 225792840, 202355008436035, 1051518440185020535448910, 6295006026005594769305465540976338825800, 250690498666352364302787619036257555981545221373940020366174361300, 76323919118339641225070197870691336391548146418602896138838604379490915124967820851616650659494440178513500

Offset: 1

Author

Darío Clavijo, Nov 04 2024

Keywords

• nonn

Comments

Number of staircase walks in a Fibonacci(n) X Fibonacci(n+1) grid where Fibonacci is A000045.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..16
• Eric Weisstein's World of Mathematics, Staircase Walk

Crossrefs

Cf. A000045, A000984 (staircase walks in a nXn grid), A001700 (staircase walks in a nX(n+1) grid).

Programs

Mathematica

Table[Binomial[Fibonacci[n+2] - 2, Fibonacci[n] - 1], {n, 12}] (* Paolo Xausa, Nov 24 2024 *)

Python

from sympy import binomial, fibonacci
a = lambda n: binomial(fibonacci(n+2)-2,fibonacci(n)-1)
print([a(n) for n in range(1, 13)])

Python

from math import comb
from gmpy2 import fib2
def A377704(n): return comb(*(lambda x:(x[0]-2,x[1]-1))(fib2(n+2))) # Chai Wah Wu, Nov 22 2024

Formula

a(n) = binomial(Fibonacci(n)+Fibonacci(n+1)-2,Fibonacci(n)-1).

a(n) = binomial(Fibonacci(n+2)-2,Fibonacci(n)-1).

a(n) >= Sum_{k=1..n-1} a(k) for n > 1.

a(n) = binomial(Fibonacci(n+2)-2,Fibonacci(n+1)-1). - Chai Wah Wu, Nov 22 2024