# Bamba amaxoxo0518

Ixoxo ligxuma lizungeze ulayini wezinombolo bese uzama ukulibamba. Ukugxuma nokubamba njalo kuyashintshana. Ixoxo liqala endaweni $$s \in \mathbb{Z}$$ futhi ngakho konke ukunyakaza ligxuma ibanga le- $$z \in \mathbb{Z}$$ (uma $$z>0$$ , ligxume ngakwesokudla, ngaphandle kwalokho uma kwesobunxele). $$z$$ kuyefana kukho konke ukweqa. Ukuqhafaza kuqukethe ukucacisa indawo ephelele. Umuntu akazi $$z$$ noma $$s$$ . Sikhombisa ukuthi kunendlela yokuhlala ubamba isele.

Okokuqala, $$a_1 = s$$ no $$a_{n+1} = a_n + z = s + n \cdot z$$ nge $$s,z \in \mathbb{Z}$$ .

Sikhetha manje

$$h:\mathbb{N} \to \mathbb{Z}^2: h(2^k r) = \left ( (-1)^{k+1} \left \lfloor \frac{k+1}{2} \right \rfloor, (-1)^{\frac{r+1}{2}} \left \lfloor \frac{r+1}{4} \right \rfloor \right )$$

njengomsebenzi onikeza (ncamashi) isiQalo senombolo sezinombolo eziphelele kuyo yonke inombolo yemvelo. Ukukhethwa kwalo msebenzi kungenxa yemisebenzi $$f(n) = (-1)^n \left \lfloor \frac{n}{2} \right \rfloor$$ , the $$\mathbb{N}$$ ku $$\mathbb{Z}$$ kanye $$g(2^kr) = (k+1, \frac{r+1}{2})$$ , okuyi- $$\mathbb{N}$$ $$\mathbb{N}^2$$ imephu nge-bijective, ekhuthazekile.

Manje sikhombisa ukukhonjiswa kwe- $$h$$ ( $$h$$ nakho kungukujova, kepha asiyidingi le ndawo).

Vumela $$(x,y) = (2^{k_1} r_1, 2^{k_2} r_2) \in\mathbb{Z}^2$$ . Kodwa ke

$$h \left ( 2^{2 \cdot 2^{k_1} r_1 - 1} \cdot (4 \cdot 2^{k_2} r_2 - 1) \right ) = (2^{k_1} r_1, 2^{k_2} r_2) = (x,y).$$

Yingakho: $$\forall (s,z) \in \mathbb{Z}^2 \, \exists \, m \in \mathbb{N}$$ nge $$h(m) = (x_m,y_m) = (s, z)$$ .

Isibonelo, uma kuyithuba lethu lokuthuthela ku- $$n = 88$$ , sibala $$h(88)=(2,3)$$ bese ukhetha $$2 + 88 \cdot 3 = 266$$ njengesikhundla.

Ngemuva kokuthi ngqo $$m$$ ihambisane $$x_m + m \cdot y_m = s + m \cdot z = a_m$$ ukhetho luwela $$x_m + m \cdot y_m = s + m \cdot z = a_m$$ .

Ngokungeziwe ku- $$h$$ , eminye imisebenzi eminingi efana nomsebenzi wokubhangqa weCantor noma i- bijective spiral iyenzeka.

Nakhu ukuqaliswa okulula kuJavaScript:

See the Pen catch the frog by David Vielhuber (@vielhuber) on CodePen.

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