155 lines
4.3 KiB
Markdown
155 lines
4.3 KiB
Markdown
---
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title: test
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subtitle:
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author: Oscar Plaisant
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documentclass: beamer
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header-includes: |
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\usepackage{amsmath, amssymb, amsfonts, mathrsfs}
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---
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#s/maths
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---
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# 1 2 4 8 ... et après ?
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---
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# Première réponse
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1, 2, 3, 4, 8, 16, 32...
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---
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# Puissances de Deux
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1, 2, 4, 8, 16, 32...
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- On multiplie par $2$ à chaque fois
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- C'est la suite des puissances de $2$
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---
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# Puissances de Deux
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1, 2, 4, 8, 16, 32...
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## Au Blackjack
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- le dé, "Videau" a ces chiffres sur ses façes
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- il permet de jouer au "quitte ou double"
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- la partie remporte 1, puis 2, puis 4 etc...
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---
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![[de Videau.png]]
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---
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# Autres réponses
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---
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# Somme des chiffres des puissances de 2
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**1, 2, 4, 8, 7, 5, 10, 11, 13**, 8, 7, 14, 19, 20, 22, 26, 25, 14, 19, 29, 31, 26, 25, 41
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| puissance de 2 | somme des chiffres |
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| -------------- | ------------------ |
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| $2^{0}=1$ | $1$ |
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| $2^1=2$ | $2$ |
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| $2^{2}=4$ | $4$ |
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| $2^{3}=8$ | $8$ |
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| $2^{4}=16$ | $7$ |
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| $2^{5}=32$ | $5$ |
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| $2^{6}=64$ | $10$ |
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| $\vdots$ | $\vdots$ |
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---
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# Somme de tous les chiffres précédents
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**1, 1, 2, 4, 8, 16, 23,** 28, 38, 49, 62, 70, 77, 91, 101, 103, 107, 115, 122, 127, 137 ...
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| nombre | somme de tous les chiffres |
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| -------- | -------------------------- |
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| $1$ | $1$ |
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| $1$ | $1+1 = 2$ |
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| $2$ | $1+1+2=4$ |
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| $4$ | $1+1+2+4=8$ |
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| $8$ | $1+1+2+4+8=16$ |
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| $16$ | $1+1+2+4+8+1+6=23$ |
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| $23$ | $1+1+2+4+8+1+6+2+3=28$ |
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| $28$ | $23+2+8 = 38$ |
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| $38$ | $38+3+8 = 49$ |
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| $\vdots$ | $\vdots$ |
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---
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# Partie entière de $\frac{n^{\;2}}{2}$
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0, 0, **2**, **4**, **8**, 12, 18, 24 ...
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$n \mapsto \left\lfloor \dfrac{n^{2}}{2} \right\rfloor$
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| $n$ | $\lfloor \frac{n^{2}}{n} \rfloor$ |
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| --- | --------------------------------- |
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| $1$ | $0$ |
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| $2$ | $2$ |
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| $3$ | $4$ |
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| $5$ | $12$ |
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| $6$ | $18$ |
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| $\vdots$ | $\vdots$ |
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---
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# U
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**1, 2, 4, 8, 9, 12, 14, 15 ...**
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les nombres qui s'écrivent avec "u"
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---
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# Arbres ternaires à n points
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![[arbres ternaires.excalidraw|700]]
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---
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# Cercles
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Nombre de régions avec $n$ cercles
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**1, 2, 4, 8**, 14, 22, 32...
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![[nombre de régions avec n+1 cercles.excalidraw|700]]
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---
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# Puissances d'une racine de 18
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**1, 2, 4, 8**, 17, 37, 76, 157...
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$n \mapsto \left\lfloor \left(\sqrt{\sqrt{18}}\right)^{n} \right\rfloor$
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$\sqrt{\sqrt{18}} \approx 2.059767144$
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---
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# Diviseurs
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Diviseurs de 88 : **1, 2, 4, 8**, 11, 22, 44, 88
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Diviseurs de 176 : **1, 2, 4, 8**, 11, 16, 22, 44, 88, 176
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Diviseurs de 352 : **1, 2, 4, 8**, 11, 16, 22, 32, 44, 88, 176, 352
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Diviseurs de 704 : **1, 2, 4, 8**, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704
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Diviseurs de 968 : **1, 2, 4, 8**, 11, 22, 44, 88, 121, 242, 484, 968
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Diviseurs de 208 : **1, 2, 4, 8**, 13, 16, 26, 52, 104, 208
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Diviseurs de 416 : **1, 2, 4, 8**, 13, 16, 26, 32, 52, 104, 208, 416
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Diviseurs de 832 : **1, 2, 4, 8**, 13, 16, 26, 32, 52, 64, 104, 208, 416, 832
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Diviseurs de 136 : **1, 2, 4, 8**, 17, 34, 68, 136.
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---
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# Polynômes de Legendre
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1, 2, 4, 8, 11 --> $-\frac{1}{6}x^{4} + \frac{11}{6}x^{3} – \frac{19}{3} x^2 + \frac{29}{3} x -4$
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1, 2, 4, 8, 12 --> $- \frac{1}{8}x^{4} + \frac{17}{12}x^{3} – \frac{39}{8}x^{2} + \frac{91}{12}x - 3$
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1, 2, 4, 8, 13 --> $- \frac{1}{12}x^{4} + x^{3} – \frac{41}{12}x^{2} + \frac{11}{2}x - 2$
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1, 2, 4, 8, 14 --> $- \frac{1}{24}x^{4} + \frac{7}{12}x^{3} – \frac{47}{24}x^{2} + \frac{41}{12}x-1$
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1, 2, 4, 8, 15 --> $\frac{1}{6}x^{3} – \frac{1}{2}x^{2} + \frac{4}{3}x+0$
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1, 2, 4, 8, 16 --> $\frac{1}{24}x^{4} – \frac{1}{4}x^{3} + \frac{23}{24}x^{2} - \frac{3}{4}x+1$
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1, 2, 4, 8, 17 --> $\frac{1}{12}x^{4} – \frac{2}{3}x^{3} + \frac{29}{12}x^{2} - \frac{17}{6}x+2$
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1, 2, 4, 8, 18 --> $\frac{1}{8}x^{4} – \frac{13}{12}x^{3} + \frac{31}{8}x^{2} - \frac{59}{12}x+3$
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1, 2, 4, 8, 19 --> $\frac{1}{6}x^{4} – \frac{3}{2}x^{3} + \frac{16}{3}x^{2} - 7x+4$
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1, 2, 4, 8, 20 --> $\frac{5}{24}x^{4} – \frac{23}{12}x^{3} + \frac{163}{24}x^{2} - \frac{109}{12}x+5$
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---
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# Il y en a plein !
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