A050613 Products of distinct terms of 1 and rest from A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^i)^bit(n,i).

1, 1, 3, 3, 7, 7, 21, 21, 47, 47, 141, 141, 329, 329, 987, 987, 2207, 2207, 6621, 6621, 15449, 15449, 46347, 46347, 103729, 103729, 311187, 311187, 726103, 726103, 2178309, 2178309, 4870847, 4870847, 14612541, 14612541, 34095929, 34095929

Offset: 0

Author

Antti Karttunen, Dec 02 1999

Keywords

• nonn

Comments

Used to produce the rows of A050609.

Also Sum(((C(2((n+((n+1) mod 2)) mod (2^floor(log_2(n)))),i) mod 2)*F(n+((n+1) mod 2)-i)),i=0..2((n+((n+1) mod 2)) mod (2^floor(log_2(n))))) or Sum(((C(2((n-(n mod 2)) mod (2^floor(log_2(n)))),i) mod 2)*L(n-(n mod 2)-i)),i=0..2((n-(n mod 2)) mod (2^floor(log_2(n))))) for all n > 1. Here F(n) and L(n) are n-th Fibonacci (A000045) and Lucas (A000032) numbers respectively.

Links

• A. Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46.

Crossrefs

Bisection: A050614.

Programs

Maple

with(combinat); A050613 := n -> product('luc(2^i)^bit_i(n,i)','i'=0..floor_log_2(n+1));
luc := n -> (fibonacci(n-1)+fibonacci(n+1));
bit_i := (n,i) -> `mod`(floor(n/(2^i)),2);
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;