A130853 Runs of 1's of lengths 1, Fibonacci numbers F(1), F(2), F(3), ... (A000045) separated by 0's.

0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Jani Melik, Jul 21 2007

Keywords

• nonn
• tabf

Comments

Might be called a Fibonacci message.

Examples

			Begin with 0. First Fibonacci number F(1)=1, so append 1's to 0 once - 01, append 0 - 010, F(2)=1, append 1's once and 0 - 01010, F(3)=2, we append two 1's and 0 - 01010110, ...
As a triangle:
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..46390
• Index entries for characteristic functions

Crossrefs

Cf. A000045, A093521, A232896 (the positions of zeros).

Cf. A001622, A003849, A005614, A194029.

Programs

Maple

ts_Finonacci_zap:=proc(n) local i,j,tren,ans; ans := [ 0 ]: for i from 1 to n do tren := combinat[fibonacci](i): for j from 1 to tren do ans:=[ op(ans), 1 ]: od: ans:=[ op(ans), 0 ]: od; RETURN(ans) end: ts_Finonacci_zap(16);
# second Maple program:
T:= n-> [0,1$(<<0|1>, <1|1>>^n)[1,2]][]:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 11 2024

Mathematica

T[n_] := Join[{0}, Table[1, MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]]]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 21 2025, after Alois P. Heinz *)

PARI

{ n=0; i=0; while(n<22, n++; i++; write("b130853.txt", i, " ", 0); k = fibonacci(n); while(k>0, i++; k--; write("b130853.txt", i, " ", 1))); }; \\ Antti Karttunen, Dec 07 2017

Formula

a(n) = b(n+1) - b(n) where b(n) = round(LambertW((phi^(3/2 + n)*log(phi))/sqrt(5)) / log(phi)), phi = (1 + sqrt(5))/2. - Alan Michael Gómez Calderón, Dec 11 2024