1175-1250
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Transcript of 1175-1250
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1175-1250
Leonardo da Pisa (noto come Fibonacci)
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Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique pariete circundatus, ut sciret, quot ex eo paria germinarentur in uno anno: cum natura eorum sit per singulum mensem aliud par germinare; et in secundo mense ab eorum natiuitate germinant. Quia suprascriptum par in primo mense germinat, duplicabis ipsum, erunt paria duo in uno mense.Ex quibus unum, scilicet primum, in secundo mense geminat; et sic sunt in secundo mense paria 3; ex quibus in uno mense duo pregnantur; et geminantur in tercio mense paria 2 coniculorum; et sic sunt paria 5 in ipso mense; ex quibus in ipso pregnantur paria 3; et sunt in quarto mense paria 8; ex quibus paria 5 geminant alia paria 5: quibus additis cum parijs 8, faciunt paria 13 in quinto mense; ex quibus paria 5, que geminata fuerunt in ipso mense, non concipiunt in ipso mense, sed alia 8 paria pregnantur; et sic sunt in sexto mense paria 21; cum quibus additis parijs 13, que geminantur in septimo†, erunt in ipso paria 34; cum quibus additis parijs 21, que geminantur in octauo mense, erunt in ipso paria 55; cum quibus additis parjis [sic] 34, que geminantur in nono mense, erunt in ipso paria 89; cum quibus additis rursum parijs 55, que geminantur in decimo mense 144; cum quibus additis rursum parijs 89, que geminantur in undecimo mense, erunt in ipso paria 233. Cum quibus etiam additis parijs 144, que geminantur in ultimo mense, erunt paria 377; et tot paria peperit suprascriptum par in prefato loco in capite unius anni. Potes enim uidere in hac margine, qualiter hoc operati fuimus, scilicet quod iunximus primum numerum cum secundo, uidelicet 1 cum 2; et secundum cum tercio; et tercium cum quarto; et quartum cum quinto, et sic deinceps, donec iunximus decimum cum undecimo, uidelicet 144 cum 233; et habuimus suprascriptorum cuniculorum summam, uidelicet 377;et sic posses facere per ordinem de infinitis numeris mensibus.
Liber Abaci, 1202,
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1 1 1 1 1 1 1 1 1 1 1 11 0 0 0 0 0 0 0 0
stringa aurea
spettro
1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22
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14930352 832040
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Massimo comun divisore con l’algoritmo di Euclide
QUANTE DIVISIONI SONO NECESSARIE ?
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al più n-1 divisioni
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1
1 2 11 3 3 1
1 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1248
16
32
64
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1
1
2
3
5
8
13
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 1
1
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64 quadretti
65 quadretti
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QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
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1101101000 in ogni prefisso il numero di 1 > del numero di 0
quante stringhe ?
ogni cammino ritorna la punto iniziale senza mai toccarlo come nodo intermedio
quanti cammini ?
quanti cammini che non toccano la diagonale ?
a+b+c+d= (a+(b+c))+d in quanti modi diversi si effettua l’operazione ?
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1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796
Numeri di Catalan
11110000 11101000 11100100 11011000 1101000
n uni e n zeri
Cn
Cn-1
in ogni prefisso il numero di 1 >= del numero di 0
in ogni prefisso il numero di 1 > del numero di 0