Pehea e helu ai i ka heluna o na paipu kila ma ka puolo hexagonal?

I ka wā e hana ai nā wili kila i kahi pūʻulu opaipu kila, pūʻolo lākou i nā ʻano hexagonal no ka maʻalahi o ka lawe ʻana a me ka helu ʻana. He ʻeono paipu i kēlā me kēia pūʻolo ma kēlā me kēia ʻaoʻao. Ehia paipu iloko o kela puolo?

Pane: 3n(n-1)+1, ʻo ia ka helu o nā paipu ma kekahi ʻaoʻao o ka hexagon maʻamau ma waho loa. 1) * 6 = 6 paipu, me 1 paipu ma waenakonu.
Ka huaʻōlelo kumu:
Paʻa nā paipu i kēlā me kēia ʻaoʻao. Aia ka papa waho loa (n-1) * 6 paipu, ka papa ʻelua (n-2) * 6 paipu, ..., ka (n-1)th papa (n-(n-1)) * 6 = 6 paipu, a ʻo ka hope 1 paipu i waenakonu. ʻO ka huina he [(n-1) + (n-2) + ... + 1]*6 + 1. ʻO ka huaʻōlelo i loko o nā pale he hōʻike ia i ka huina o ka pūʻulu helu (ka pūʻulu o nā huaʻōlelo mua a me hope i puʻunaue ʻia me 2, a laila hoʻonui ʻia me ka n-1 e loaʻa ai ka n*(n-1)/2).
Loaʻa kēia i ka 3n*(n-1)+1.