Beware of different golden ratio symbols used by different authors!

At this web site Phi is 1.618033... and phi is 0.618033.. but Vajda(see below) and Dunlap(see below) use a symbol for -0.618033.. .

Where a formula below (or a simple re-arrangement of it) occurs in either Vajda or Dunlap's book, the reference number they use is given. Dunlap's formulae are listed in his Appendix A3. Hoggatt's formula are from his "Fibonacci and Lucas Numbers" booklet. Full bibliographic details are at the end of this page.

As used here

Vajda Dunlap Description floor(x)

the nearest integer < x.

When x>0, this is "the integer part of x" or "truncate x" i.e. delete any fractional part after the decimal point.

3=floor(3)=floor(3.1)=floor(3.9), -4=floor(-4)=floor(-3.1)=floor(-3.9)

round(x)

the nearest integer to x, equivalent to trunc(x+0.5) 3=round(3)=round(3.1), 4=round(3.9), -4=round(-4)=round(-3.9), -3=round(-3.1) 4=round(3.5), -3=round(-3.5)

ceil(x)

the nearest integer > x.

3=ceil(3), 4=ceil(3.1)=ceil(3.9), -3=ceil(-3)=ceil(-3.1)=ceil(-3.9)

Cr; n choose r, the element in row n nwr>

column r of Pascal's Triangle, the coefficient of xr in (1+x)n, the number of ways of choosing r objects from a set of n different objects. n>0 and r>0.

F(i) is the Fibonacci series and L(i) is the Lucas series.

Formula |
Vajda |
Dunlap |
Comments |

F(0) = 0, F(1) = 1, F(n+2) = F(n + 1) + F(n) |
- |
- |
Fibonacci series |

L(0) = 2, L(1) = 1, L(n + 2) = L(n + 1) + L(n) |
- |
- |
Lucas series |

G(n + 2) = G(n + 1) + G(n) |
3 |
4 |
Generalised Fibonacci series, G(0) and G(1) needed |

Lj |
=T |
=T,63 |
Phi and -phi are the roots of x2 = x + 1 |

Phl = 2 |
= -CT |
= -0,65 |
Dunlap occasionally uses 0 to represent our phi = 0.61803.., but more frequently he uses 0 to represent -0.618033..! |

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