A New Integer/Number Sequence Generator Dann Passoja Winter 016
Background and Interest:... The Basis... 3 In a Broad Perspective (Philosophy)... 3 My Discovery... 4 Number Series... 4 Series of Partial Sums... 4 Geometric Series... 4 Series that converges having alternating sign... 5 Integer Sequences... 5 Lucas Series... 5 The Equations One and Two In More General Terms... 6 Quantum Mechanics and the Creation Annihilation Operator... 6 p m + 1 mω x = E = hω n + 1... 6 The Generalized Sequence-Series Generator... 7 The Analytical Method... 8 The Analytical Flow Diagram... 10 Results of Analyses for 0 Different Integer Sequences -Power Is a Variable Used to Generate These Sequences... 11 Results of Analyses of Six Different Integer Sequences- Power is Constant and N is Variable to Generate these Sequences... 1 Number Sequences Made from the C/A Equation... 13 Semi Log Plot of QM data shown above... 14 Summary and Conclusions... 14 Background and Interest: Over a period of about 0 years I ve maintained an interest in numbers, in fact, certain numbers have fascinated me and have had significant influences other people as well. This fascination has been carried through the ages stemming from the times of the Grecian Empire from approximately 000BC. There are two numbers that are stand outs-golden Ratio and the Silver Ratio.
The book by Huntley The Divine Proportion whet my appetite for Phi 1+ 5 and in working with the Passoja Lakahtakia Set which is based on the Delannoy numbers I happened on a relation for σ = 1+. Both of these numbers have the extraordinary property of 1 φ = 1.618033... φ = 0.618033... 1 σ =.4141... σ = 0.4141... They both have unique properties with respect to their inverse namely, they are equal to their decimal fractions. The Basis It s not very complicated, nor is it very difficult to understand. The two numbers, ϕ and σ can be found by solving a simple equation (below). It s this this equation that forms the basis for the sequence generator. These two form a starting point for this work. A statement of the basis sets the tone for this work. They re elementary quadratic equations in x: for φ x + 1 x = 1 (1 for σ x + 1 x = ( In a Broad Perspective (Philosophy) However if truth be known these two numbers are just a starting point for me because I envision numbers systems in different terms than other people do. My thoughts lie along the lines of the mathematician L.Kroneker when he said.god invented the integers all the rest are the works of man It s not too difficult to understand this if you re a mathematician or a scientist, but otherwise, the integers are taken for granted and devalued and because everyone uses them they aren t worth very much. Unless, of course, we didn t have any. 3
My Discovery Integer Sequences- Number Series I have discovered a generator for creating sequences of integers and numbers. By using this generator it s possible to make a sequence of integers, for example, 7,14, 3,34,47,6 etc which have important, special, mathematical properties. It can also be used to create series of numbers that are fractions and to analyze the relationships between them. Number Series I d be remiss if I didn t introduce series in general and number sequences in particular. In order to form a coherent base for this work, I I introduce several of them: Series of Partial Sums Geometric Series 4
Series that converges having alternating sign Integer Sequences There are series that are called integer Number Sequences. The most famous of these is the Fibonacci series as well as others that are shown below: Fibonacci series Which is constructed in the following manner: starting with 1,1 add the leading 1 to the 1 behind it making, then take the and add it to the 1 behind it etc. Phi is related to the ratio: Where F n+1 n+1 th Fibonacci number in the series. Lucas Series is like the Fibonacci series but has different starting conditions. 5
The Equations One and Two In More General Terms Equations 1 and were important because of their simplicity and because of their relationship to φ and σ. However in this form their applications are limited because they are narrowly focused. I wanted to create an equation that would cover a broader base that would cover the mathematical landscape that I had in mind.. Quantum Mechanics and the Creation Annihilation Operator The Creation/Annihilation operator is an extremely useful tool in quantum physics. It has become useful in studies done on the harmonic oscillator and quantum field theory. It is a representation of the Hamiltonian and with a little manipulation the Schroedinger equation can be derived from it. In working through some of the quantum mechanical identities the C/A operator is basically a quadratic representation of x and its inverse. Its development follows: Writing the Hamiltonian for the harmonic oscillator: p m + 1 mω x = E = hω n + 1 then using Debroglie equation which I bend the rules a little in order to get a x and its inverse simply into the equations: p = h x p m + 1 mω x = E = hω n + 1 h mx + 1 mω x = hω n + 1 h mx hω + 1 mω x hω x q x x + 1 x = n + 1 q let x q = 1 then 1 (3 6
x + 1 x = n + 1 Here is the C/A equation in terms of x and n. Solving for for x: lim x n =lim n x = ± n + 5 ± n 3 4 n + 5 + n 3 4 = 7 + i 4 = 3 4 (4 The Generalized Sequence-Series Generator The properties of the above equations showing a linear term and its inverse will be explored next. However, the C/A operator brings up the importance of including quadratic and an inverse quadratic term in this analysis. However, an even more general relationship will be used; one that encompasses the linear, quadratic and more. For this reason the scope of this study will include terms to the I th power: Linear x + 1 x = n Quadratic x + 1 x = n Powers of i x i + 1 x i = n thus: i = 1,,3,4,... n = 1,,3,4... 7
Getting the Terms Straight Variable I=An integer having no decimal fraction, c is computed from it i c n coefficients used for calculating the terms in series Source Fundamental Assumption An intermediate parameter I an integer- power A variable integer value 1,,3! A number that results from solving the fundamental equation, can have a variable I or a fixed I and variable c-s x an unknown Just an unknown The Analytical Method The operational computations are as follows: this is an example of a calculation for n=3 Example x + 1 x = 3 x = ± 5 ± 3 c 3 i = c 3 + 1 c 3 i = a 3,b 3,c 3...an int eger sequence 8
These are the values of the terms that were used for input into the sequence generators N N- Input Coefficients 0 0 1 1 3 ± 5 ± 3 4 ± 3 ± 6 ± ± 3 7 ±3 5 ± 7 8 ± 15 ± 4 9 ± 77 + 9 10 ± 6 ± 5 ( ) 9
The Analytical Flow Diagram The flow diagram shown below is a succinct summary of the computational scheme that was used in this work. Sequence/Series Generation Includes square roots and approximate numbers but no functions Series Integer Sequences Only Integers Choose Integer Choose Integer Generator Solve Quadratic Equation for x Put Solution Back Into Generator Keep i constant use solutions of the quadratic equation as variables Keep solution of the quadratic equation as a constant use i as a variable Integer Sequences Number Series Integer Sequences i identifies the sequence Number Series Unidentified Sequences Unidentified Series Lucas Sequences Unidentified Sequences Unidentified Series 10
Results of Analyses for 0 Different Integer Sequences -Power Is a Variable Used to Generate These Sequences 10,10,98,970,960,95050,940898... OEIS087799 a(n)=10a(n-1)-a(n-) 9,9,79,70,639,55449... OEISA056918 a(n)=9a(n-1)-a9n-) 8,8,6,488,384,3048,3814,1874888... 7 7,47,3,07,1517,10368... OEIS!086903 a(n)=*a(n-1)-a(n-) 6,6,34,198,1154,676 OEISA056854 a(n)=lucas(4n) 5,5,3,110,57,55... OEISA003499 a(0)=,a(1)=6: for n>=,a(n)=6a(n-1)-a(n-) 4,4,14,5,194,74... OEISA003501 a(n)=5a(n-1)-a(n-) 3,3,7,18,47,13,3... OEIS003500 a(n)=4a(n-1)-a(n-) 1 1,-1,-,-1,1... OEISA00548 Bisection of Lucas Numbers a(n)=l(n) 0 Sequences/Series, 0,-,0,,0,-,0,-... 11,11,119,198,14159,154451,168480 OEISA057076 Chebyshev Fibonacci (Generalized) 1,1,169,016,405,8686 OEISA087800 a(n)=1a(n-1)-a(n-) 13 14,13,167,158,7887,360373,465696, OEISA078363 Chebychev T sequence with Diophantine Property 14,194,70,37634,54174,730080, OEISA06790 a(n)=14a(n-1)-a(n-) 15 15,3,3330,4977... OEIS78365 Chebychev T Sequence 16,16,54,4048,64514,108176,1638630... OEISA09077 a)n)=18a(n-1)-a(n-) 17 18 17,31,49,71,37...,18,3,5778,10368,1860498,... OEIS0560 OEISA08715 a(n)=n^-1 Lucas(6n) 18a(n-1)-a(n-) 19,19,359,680,18879,441899 OEIS078369 Chebychev 0 0,7940,3160100,63043598,15,7711860 OEISA09078 a(n)=0a(n-1)-a(n-) 11
Results of Analyses of Six Different Integer Sequences- Power is Constant and N is Variable to Generate these Sequences 3,70,1098,390,10368,3814,4980? 6 13,74,55,676,1517,3048 OEISA30586 a(n)=n^5-5n^3+5n 5 4 47,194,57,1154,07,384,639 Sequence is n^4_8n^3+0n^+16n+ 3 18,5,110,198,3,488 OEISA058794 3,7,14,3,34,47,6...? Array (a frieze pattern) defined by a(n,k)=(a(n-1,k)a(n-1,k+1-1)/a(n-,k+1) read by anti diagonals 1,3,4,5,6,7... Sequences/Series Fixed i Variable N 1
Number Sequences Made from the C/A Equation Variable i.1041 3.0035 5.186.474 3.894 7.011 1 3 4 5 6 Quantum Mechanical C/A Number Sequence 1 3 4 5 6 1.0473 1.56617 1.968 0.75181 1.34409 1.75483 Variable n 13
Semi Log Plot of QM data shown above The correlation coefficient is 1.0 The equation for the curve is semi-log C / A = e 0.306n This is a just a demonstration of how real numbers have to be analyzed, that is They re not integers so what do I do with them? Summary and Conclusions I ve discovered a way to compute integer sequences by performing computations by means of mathematical operations. The method is based on an integer/number sequence generator by which a great many integer sequences can be determined. All but one (out of 6 that were tried) were found on line The On-Line Encyclopedia of Integer Sequences or OEIS. 14
The use of the generator is far from exhausted, on the contrary, the sequences in this paper were done as a warm up in order to see How everything worked and to determine Where the bugs were. I m happy to say that everything went without a problem with all of my time being spent learning how everything worked. It was a very satisfying experience. I would like to thank the people at OEIS for what they have done. It has been an enormous help in making this work come to life. It was possible to make calculations of a sequence, let s say, and then find it s pedigree on OEIS. There were some sequences that were not on OEIS and I reflected on this a bit. With the success of the sequence generator I could tell that these sequences should have been on the OEIS list but they were not! They were missing. Others that couldn t be found, just couldn t be found, however I began to understand that they were perfectly good sequences but hadn t been discovered yet. All in all working with the sequence generator has been a delightful experience. Dann Passoja and Family New York, New York February 016 15