Quantum coin flipping, qubit measurement, and generalized Fibonacci numbers

The problem of Hadamard quantum coin measurement in $n$ trials, with arbitrary number repeated consecutive last states is formulated terms Fibonacci sequences for duplicated states, Tribonacci numbers triplicated and $N$-Bonacci $N$-plicated states. probability formulas position are derived Lucas numbers. For generic qubit coin, the expressed by more general, polynomials probabilities. generating function probabilities, Golden Ratio limit these probabilities Shannon entropy corresponding determined. By generalized Born rule universality $n$-qubit gate, we formulate construct projection operators Hilbert space, constrained on tree results to qutrit qudit coins, described Fibonacci-$N$-Bonacci sequences.