Fibonacci numbers

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

Fibonacci numbers, where the number sought is the nth number in a series defined in terms of relationships to the n-1, n-2, etc. members of the series. All recursive procedures must have a terminating condition, so that they do not call themselves endlessly. 

PHI is the ratio of any two sequential numbers in the Fibonacci (V sequence. If you take the numbers 0 and 1, then create each subsequent number in the sequence by adding the previous two numbers, you get the Fibonacci sequence. For example, 0, 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Ifyou sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times (10) the next Fibonacci number. This property results in the Fibonacci spiral seen in everything from seashells to galaxies, and is written mathematically as l2 + l2 + 22 + 32 + 52 = 5 x 8. 

Fig. 3. All five Clar structures of a branched benzenoid hydrocarbon and the corresponding five colored Gutman trees. The indicated coloring generates F4, the fourth Fibonacci number...

The rational fractions defined by successive Fibonacci numbers in the sequence ... 

Figure 8-12. (a) The scattered leaf arrangement (phyllotaxis) of Plantago media (drawing by Ferenc Lantos, Pecs). Fibonacci numbers of spirals in the patterns of (b) Scales of a pinecone and (c) A cactus in Hawaii (b and c, photographs by the authors). 

Fig. 20, All unbranched catacondensed benzenoids with 2-segments only (non-helicenic fibonacenes) for h < 8. K numbers are given they are the Fibonacci numbers Fh + l F0 = Ft = 1)...

The stability problem is solved on noting that allowed fractions at small atomic number begin at unity and approach r with increasing Z. This trend should, by definition follow Farey fractions determined by Fibonacci numbers. The first few Fibonacci numbers are 0,1,1,2,3,5,8,13,21, etc. The ra-... 

Because the range of nuclidic stability is bounded by fractions that derive from Fibonacci numbers, it probably means that nuclear stability relates directly to the golden mean. To demonstrate this relationship it is noted that the plot of A vs Z, shown in figure 13 for the A(mod4) = 0 series of nuclides, separates into linear sections of constant neutron excess (A — 2Z) and slope 2. Each section terminates at both ends in a radioactive nuclide. The range of stability for each section follows directly from... 

Fibonacci numbers [44] K values of the zigzag benzenoid hydrocarbons benzene, nai thalene, phenanthrene, chrysene,. ... 

One recalls the above equation as the Binet formula of Fibonacci numbers [44]. Furthermore, for such a member of homologous set, the number of selections of k resonant hexagons is given by [34,44]... 

A.N. Philippou, G.E. Bergum and A.F. Horadam, eds-. The Fibonacci Numbers. Applications of Fibonacci Numbers in the... 

Keifer (Kl) and Johnson (J2) have shown that there is another sequential method that is even better. It involves using Fibonacci numbers F , which are defined inductively as follows ... 

Thus each Fibonacci number is the sum of the two preceding it. The Fibonacci numbers associated with the first eleven integers are given in the following table. 

Fibonacci numbers -> symmetry descriptors (O Merrifield-Simmons index) fieid effect -> electronic substituent constants fieid-fitting alignment -> aiignment ruies... 

Hosoya found that for linear graphs the Z indices are the Fibonacci numbers, i.e. Z = Ea) where A is the number of atoms in the molecular graph therefore, for a linear graph, it is closely related to the -> Merrifield-Simmons index [Gutman et ai, 1992 Randic et ai, 1996b]. 

This recursion formula is identical to the definition of Fibonacci numbers ... 

Moreover, again for linear graphs (i.e. n-alkanes), the Hosoya Z index coincides with the Fibonacci number F , and thus Hosoya and Merrifield-Simmons indices are both closely and directly related. For monocyclic graphs C and iso-path graphs i - L the following relationships between Fibonacci numbers and the Merrifield-Simmons index hold ... 

Table S-3 collects Fibonacci numbers and the corresponding Merrifield-Simmons indices for number n of vertices between 0 -16.

Table S-3. Fibonacci numbers (F ) and corresponding Merrifield-Simmons indices for linear (Z- ), cyclic (C), and iso-path graphs...

Balaban, A.T. and Tomescu, T. (1985). Chemical Graphs. XLI. Numbers of Conjugated Circuits and Kekule Structures for Zigzag Catafusenes and (j, k)-hexes Generalized Fibonacci Numbers. MATCH (Comm.Math.Comp.Chem.), 17, 91-120. 

Randic, M., Morales, D.A. and Araujo, O. (1996b). Higher-Order Fibonacci Numbers. J.Math. Chem., 20, 79-94. 

The symmetry of quasicrystals can be represented by introducing a different perturbation function, which is based on the Fibonacci numbers. An infinite Fibonacci sequence is derived fi-om two numbers, 0 and 1, and is formed according to the following rule ... 

The smdy of the activities of some homologous compounds, can, through interpolation, identify which term is associated with the highest potency. The optimization method proposed by Bustard, makes use of the Fibonacci numbers, and allows the identification of the most active compound (presumed to exist in a given interval) with the smallest possible number of syntheses (see also Refs. [14,15]). 

Fibonacci numbers symmetry descriptors (0 Merrifield-Simmons index)... 

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