Reproduced and discussed in Curchin & Herz-Fischler and discussed in Herz-Fischler's book, pp. Marginal note to II.11 in Luca Pacioli's copy of his 1509 edition of Euclid. 2-3.) I have colour slides of this from L.IV.20 & 21 and Conventi Soppresi, C. (English translation in: Struik, Source Book, pp. Rabbit problem - the pair propagate in the first month so there are Fn+2 pairs at the end of the n-th month. 283-284 (S: 404-405): Quot paria coniculorum in uno anno ex uno pario germinentur. Ron Knott has a huge website on Fibonacci numbers and their applications, with material on many related topics, e.g. Claims to be 'the first attempt to compile a definitive history and authoritative analysis' of the Fibonacci numbers, but the history is generally second-hand and marred with a substantial number of errors, The mathematical work is extensive, covering many topics not organised before, and is better done, but there are more errors than one would like. Fibonacci and Lucas Numbers with Applications.
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He also discusses the origin of the term 'golden section', sketching the results given in Herz-Fischler's book. SOURCES - page 2 Pyramid, the Parthenon, Renaissance paintings and/or the human body and that the Golden Rectangle is the most pleasing - with 59 references. This surveys many of the common misconceptions - e.g. 157-162 discuss early work relating the Fibonacci sequence to division in extreme and mean ratio. Retitled: A Mathematical History of the Golden Number, with new preface and corrections and additions, Dover, 1998. Wilfrid Laurier University Press, Waterloo, Ontario, 1987. A Mathematical History of Division in Extreme and Mean Ratio. Pacioli and Kepler, described below, seem to be the first to find this. Discusses the history of the result that the ratio Fn+1/Fn approaches φ. De quand date le premier rapprochement entre la suite de Fibonacci et la division en extrême et moyenne raison? Centaurus 28 (1985) 129-138. Brief sketch, with lots of typographical errors. Fibonacci numbers: Their history through 1900. 11: The golden section and phyllotaxis, pp. The golden section, phyllotaxis, and Wythoff's game.
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Tell me, O learned man, the number of progeny produced during twenty years by one cow." WESTERN HISTORIES H. The calves become young and themselves begin giving birth to calves when they are three years old. Dvivedi, Indian Press, Benares, 1942), ?NYS - quoted by Kripa Shankar Shukla in the Introduction to his edition of: Narayana Pandita (= Nārāyaņa Paņdita) Bījagaņitāvatamsa Part I Akhila Bharatiya Sanskrit Parishad, Lucknow, 1970, p. + xq-1)p and relates to ordered partitions using 1, 2.
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He gives rules which are equivalent to finding the coefficients of (1 + x +. 13, where each term is the sum of the last q terms. Narayana Pandita (= Nārāyaņa Paņdita’s ) Gaņita Kaumudī (1356) studies additive sequences in chap. It also gives the relation Fn+1 = Σi BC(n-i,i). The Prākŗta Paińgala (c1315) gives rules for finding the k-th sequence of weight n and for finding the position of a particular sequence in the list of sequences of weight n and the positions of those sequences having a given number of 2s (and hence a given number of letters). next." This is repeated by later authors. Hemacandra (c1150) states "Sum of the last and last but one numbers. Gopāla (c1134) gives a commentary on Virahāńka which explicitly gives the numbers as 3, 5, 8, 13, 21.
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Virahāńka (c7C) is slightly more explicit. Pińgala (c-450) studied prosody and gives cryptic rules which have been interpreted as methods for generating the next set of sequences, either classified by number of letters or by weight and several later writers have given similar rules. Classifying by weight gives the number of sequences of 1s and 2s which add to the weight n and this is Fn+1. In early Indian poetry, letters had weights of 1 or 2 and meters were classified both by the number of letters and by the weight. The so-called Fibonacci numbers in ancient and medieval India. We use the standard form: F0 = 0, F1 = 1, Fn+1 = Fn + Fn-1, with the auxiliary Lucas numbers being given by: L0 = 2, L1 = 1, Ln+1 = Ln + Ln-1. ARITHMETIC & NUMBER-THEORETIC RECREATIONS 7.A.