A377704 - OEIS

%I #24 Nov 26 2024 08:42:00

%S 1,1,3,15,330,50388,225792840,202355008436035,

%T 1051518440185020535448910,6295006026005594769305465540976338825800,

%U 250690498666352364302787619036257555981545221373940020366174361300,76323919118339641225070197870691336391548146418602896138838604379490915124967820851616650659494440178513500

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

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

%H Paolo Xausa, <a href="/A377704/b377704.txt">Table of n, a(n) for n = 1..16</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StaircaseWalk.html">Staircase Walk</a>

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

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

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

%F a(n) = binomial(Fibonacci(n+2)-2,Fibonacci(n+1)-1). - _Chai Wah Wu_, Nov 22 2024

%t Table[Binomial[Fibonacci[n+2] - 2, Fibonacci[n] - 1], {n, 12}] (* _Paolo Xausa_, Nov 24 2024 *)

%o (Python)

%o from sympy import binomial, fibonacci

%o a = lambda n: binomial(fibonacci(n+2)-2,fibonacci(n)-1)

%o print([a(n) for n in range(1, 13)])

%o (Python)

%o from math import comb

%o from gmpy2 import fib2

%o def A377704(n): return comb(*(lambda x:(x[0]-2,x[1]-1))(fib2(n+2))) # _Chai Wah Wu_, Nov 22 2024

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

%K nonn

%O 1,3

%A _Darío Clavijo_, Nov 04 2024