Pascal's triangles for Fibonacci and Lucas numbers (Slider version 2.0)
修正版パスカルの三角形によるリュカ数列とフィボナッチ数列 (スライダー・バージョン 2.0)

Lucas numbers and Fibonacci numbers
Fig. 112 Harmonies and symmetries about Newton's binomial theorem, applied Pascal's triangle, and generalized Fibonacci sequences with the line using zero (2023)

Fig. 113 Harmonies and symmetries about Newton's binomial theorem, applied Pascal's triangle, and Lucas sequences with the line using powers of two (2023)

Fig. 114 Beauty and truth of visualizations between generalized Fibonacci sequence and Lucas sequence using generalized binomial theorem (2023)

Fig. 115 Beauty and truth of visualizations between generalized Fibonacci sequence and Lucas sequence using generalized binomial theorem (Part 2)(2023)

Fig. 116 Visualizations about equiangular spirals with generalized Fibonacci sequence and Lucas sequence using Kepler triangles,similar metallic ratios and Pythagorean theorem (2023)

Fig. 117 Visualizations about equiangular spirals with generalized Fibonacci sequence and Lucas sequence using isosceles right triangles, similar metallic ratios and Pythagorean theorem (2023)

Fig. 118 Visualizations about equiangular spirals with generalized Fibonacci sequence and Lucas sequence using related right triangles, similar metallic ratios and Pythagorean theorem (2023)

Fig. 119 Visualizations of golden ratio using nested radical, extended continued fraction, Fibonacci sequence, Lucas sequence, squares, and circles (2023)

Fig. 120 Visualizations of silver ratio using nested radical, extended continued fraction, Pell sequence, Pell-Lucas sequence, squares, and circles (2023)

Fig. 121 Visualizations of first and second ratio using Fibonaccisequence, Lucas sequence, Jacobstahl sequence, Jacobstahl-Lucas sequence, and circles (2023)

Fig. 122 Extended version of Fibonacci sequences triangleand Lucas sequences triangle using the concept of Hosoya's traiangle (2023)

Fig. 123 Visualization about Padovan sequence using Pascal's traiangle (2023)

Fig. 124 Visualization about Perrin sequence using modified Pascal's traiangle (2023)

Fig. 125 Visualization about Padovan and Perrin sequence using modified Pascal's traiangles with plastic ratio (2023)

Fig. 126 Visualization about Padovan sequence and its spiral using modified Pascal's traiangles with plastic ratio (2023)

Fig. 127 Visualization about golden ratio or plastic ratio and these related sequences using modified Pascal's traiangles or equiangular spirals (2023)

Fig. 128 Visualization about Padovan sequence and Perrin sequence with plastic ratio(2023)

Fig. 129 Visualizations about equiangular spirals with Fibonacci sequence and Lucas sequence using Kepler triangles,and Pythagorean theorem (Part2, 2023)

Fig. 130 Squares and circles about golden ratio and silver ratio with Fibonacci sequence, Lucas sequence, Pell sequence, Pell Lucas sequence, and extened version of continued fraction and nested radical of metallic raios (2023)

Fig. 131 Squares and circles about golden ratio and silver ratio with Fibonacci sequence, Lucas sequence, Pell sequence, Pell Lucas sequence, and extened version of continued fraction and nested radical of metallic raios (Part2, 2023)

Fig. 132 Type k-2 skkiped Fibonacci sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 133 Type k-2 skkiped Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 134 Concept of the one skkiped generalizecd Fibonacci sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 135 Concept of the two skkiped generalizecd Fibonacci sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 136 Concept of the one skkiped generalizecd Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 137 Concept of the two skkiped generalizecd Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 138 Concept of the two skkiped generalizecd Fibonacci sequence and Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 139 Concept of the one or two skkiped Pell sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 140 Concept of the one or two skkiped Pell Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 141 Concept of the one skkiped generaluzed Fibonacci sequence with modified Pascal's triangles and similar metallic ratios (2023)

Fig. 142 Concept of the one or two skkiped Jacobsthal sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 143 Concept of the one or two skkiped Jacobsthal Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 144 Extended nested radical about first and second ratios of similar metallic ratios (2023)

Fig. 145 Extended nested radical about golden and silver ratios of metallic ratios (2023)

Fig. 146 Equilateral triangles using Fibonacci and Lucas, or Pell and Pell Lucas sequenceses (2023)

Fig. 147 Equilateral triangles using Fibonacci and Lucas sequenceses (2023)

Fig. 148 Equilateral triangles using Fibonacci and Lucas, or Pell and Pell Lucas sequenceses (Part2, 2023)

Fig. 149 The golden ratio with Pythagorean theorem, Fibonacci sequence, Kepler triangle, equiangular spiral, and nested radicals (2023)

Fig. 150 Visualization of the ratio such as golden ratio and plastic ratio with the related sequences (2023)

Fig. 151 Spirals using equilateral triangles such as plastic ratio with the related sequences (2024)

Fig. 152 Spirals using equilateral triangles such as plastic ratio with the related sequences (Part 2, 2024)

Fig. 153 Spirals using equilateral triangles such as plastic ratio with the related sequences (Part 3, 2024)

Fig. 154 Spirals using equilateral triangles such as golden ratio with the Fibonacci sequence, Lucas sequence, and Mulatu sequence (2024)

Fig. 155 Spirals using equilateral triangles such as golden ratio, silver ratio, bronze ratio with the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, k Pell sequence, and k Pell-Lucas sequence (2024)

Fig. 156 Dynamically modified pascal's triangles for Fibonacci sequences and Lucas sequences (2023)

Fig. 157 Dynamically modified pascal's triangles for 0, 1, and 2 skipped Fibonacci sequences (2023)

Fig. 158 Dynamically modified pascal's triangles for 0, 1, and 2 skipped Lucas sequences (2023)

Fig. 159 Dynamically modified pascal's triangles for 2 skipped Gibonacci sequences (2023)

Fig. 160 Dynamically modified pascal's triangles for weighted Fibonacci sequences and Lucas sequences (2023)

Fig. 161 Concepts of plastic ratios and the related sequences with equilateral triangles (2024)

Fig. 162 Dynamically modified pascal's triangles for Padovan sequences and Perrin sequences (2023)

Fig. 163 Dynamically modified pascal's triangles for Padovan sequences and Perrin sequences (Part 2, 2023)

Fig. 164 Visualization of Matrices for Padovan sequences and these computing (2024)

Fig. 165 Visualization of Matrices and these computing for skipped Fibonacci sequences (2024)

Fig. 166 Visualization modified Pascal's triangles for 1 or 2 skipped Fibonacci sequences and Lucas sequences (2023)

Fig. 167 Visualization modified Pascal's triangles for 1 or 2 skipped Pell sequences and Pell-Lucas sequences (2023)

Fig. 168 Visualization modified Pascal's triangles for 1 or 2 skipped Jacobsthal sequences and Jacobsthal-Lucas sequences (2023)

Fig. 169 Concepts of modified Pascal's matrices and the related metallic ratios (2024)

Fig. 170 Numerical examples of modified Pascal's matrices and the related metallic ratios (2024)

Fig. 171 Numerical examples of modified Pascal's matrices for 1 skipped Fibonacci, Pell, Jacobstahl, Lucas, Pell-Lucas, and Jacobstahl-Lucas sequences (2024)

Fig. 172 Numerical examples of modified Pascal's matrices for 2 skipped Fibonacci, Pell, Jacobstahl, Lucas, Pell-Lucas, and Jacobstahl-Lucas sequences (2024)

Fig. 173 Numerical examples of modified Pascal's matrices for negativeorders of skipped Fibonacci sequence (2024)

Fig. 174 Numerical examples of modified Pascal's matrices for negativeorders of skipped Lucas sequence and Jacobstahl-Lucas sequence (2024)

Fig. 175 GIF Animation of modified Pascal's triangles for several skipped Fibonacci, Pell, Jacostahl sequences and these related Lucas sequencea (2023)

Fig. 176 Double spirals using equilateral triangles such as golden ratio, silver ratio, bronze ratio with the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, k Pell sequence, and k Pell-Lucas sequence (2024)

Fig. 177 Triple spirals using equilateral triangles for x powers of 2 which is the sequence basad on Jacobsthal sequence with changing initial values (2024)

Fig. 178 Single, double, and triple spirals using equilateral triangles using plastic ratio, golden ratiom and 2 or these related sequences spirals such as Padovan sequence, Fibonacci sequence, and Jacobsthal sequence with changing initial values (2024)

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