A048757 - OEIS

A048757

Sum_{i=0..2n} (C(2n,i) mod 2)*Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+2).

1, 4, 9, 33, 56, 203, 441, 1596, 2585, 9353, 20304, 73461, 124033, 448756, 974169, 3524577, 5702888, 20633243, 44791065, 162055596, 273617239, 989956471, 2149017696, 7775219067, 12591974497, 45558191716, 98898651657

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The history of 1-D CA Rule 90 starting from the seed pattern 1 interpreted as Zeckendorffian expansion.

Also, product of distinct terms of A001566 and appropriate Fibonacci or Lucas numbers: a(n) = FL(n+2)Product(L(2^i)^bit(n,i),i=0..) Here L(2^i) = A001566 and FL(n) = n-th Fibonacci number if n even, n-th Lucas number if n odd. bit(n,i) is the i-th digit (0 or 1) in the binary expansion of n, with the least significant digit being bit(n,0).

LINKS

Table of n, a(n) for n=0..26.

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

EXAMPLE

1 = Fib(2) = 1;

101 = Fib(4) + Fib(2) = 3 + 1 = 4;

10001 = Fib(6) + Fib(2) = 8 + 1 = 9;

1010101 = Fib(8) + Fib(6) + Fib(4) + Fib(2) = 21 + 8 + 3 + 1 = 33; etc.

MATHEMATICA

Table[Sum[Mod[Binomial[2n, i], 2] Fibonacci[i + 2], {i, 0, 2n}], {n, 0, 19}] (* Alonso del Arte, Apr 27 2014 *)

CROSSREFS

a(n) = A022290(A038183(n)) = A022290(A048723(5, n)) = A003622(A051656(n)) = A075148(n, 2)*A050613(n). Third row of A050609, third column of A050610.