FIBONACCI Y EL NUMERO AUREO EN EL PRISMA COMBINATORIO ∆𝒌
∆𝟎
(𝒇𝒌𝒏−𝟐 + 𝒇𝒌𝒏−𝟏 ) + 𝒇𝒌−𝟏 = 𝒇𝒌𝒏 𝒏 ф𝒌𝒏 =
𝒏+𝒌−𝟏−𝒊 𝒏−𝟏−𝒊 )( ) 𝒌 𝒊 (𝒏−𝟐)/𝟐 𝒏+𝒌−𝟐−𝒊 𝒏−𝟐−𝒊 ∑𝒊=𝟎 ( )( ) 𝒊 𝒌 (𝒏−𝟐)/𝟐
𝒇𝒌𝒏 𝒇𝒌𝒏−𝟏
=
∑𝒊=𝟎
𝝓𝒌 = 𝒍𝒊𝒎 𝒏→∞
(
𝒇𝒌𝒏+𝟏 𝒇𝒌𝒏
=𝝓
𝒙
𝑭𝒌 (𝒙) = (𝟏−𝒙−𝒙𝟐 )𝒌+𝟏 𝒇𝒌𝒏 =
𝟏 √𝟓
∑𝒏𝒊=𝟏 𝒇𝒌−𝟏 (𝝓𝒏−𝒊+𝟏 − 𝝋𝒏−𝒊+𝟏 ) 𝒊
(𝒇𝟎𝒏−𝟐 + 𝒇𝟎𝒏−𝟏 ) = 𝒇𝟎𝒏 𝒏−𝟏−𝒊 ) 𝒊 (𝒏−𝟐)/𝟐 𝒏−𝟐−𝒊 ∑𝒊=𝟎 ( ) 𝒊 (𝒏−𝟐)/𝟐
𝒇𝟎
ф𝟎𝒏 = 𝒇𝟎 𝒏 = 𝒏−𝟏
𝝓 = 𝒍𝒊𝒎
∑𝒊=𝟎
(
𝒇𝟎𝒏+𝟏
𝟎 𝒏→∞ 𝒇𝒏
𝒙
𝑭𝟎 (𝒙) = 𝟏−𝒙−𝒙𝟐 𝒇𝟎𝒏 =
𝟏 √𝟓
(𝝓𝒏 − 𝝋𝒏 )
Enrique R. Acosta R. 2019