ポッキー数列の加法定理 / Pocky number additon theorem

Transcript

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2. $POUFOUT
   
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   (2) What is Pocky (Repunit) number ?
   
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3. 5PEBZ /PWJT1PDLZ EBZ
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   sin + = sin cos + cos sin
   cos + = cos cos − sin sin
   3
   
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4. 'JCPOBDDJOVNCFS
   ,-.
   = ,-/
   + ,
   , 1
   = 0, /
   = 1, .
   = 1, ⋯
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   5-6
   = 5-/
   6
   + 5
   67/
   &Y8
   = /-9
   = /
   :
   + .
   9
   = 1 ; 2 + 1 ; 3 = 5
   4
   
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5. 1PDLZ(Repunit) OVNCFS
   @A/1,,
   = ,
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   1
   = 0, /
   = 1, .
   = 11, :
   = 111, 9
   = 1111,
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   5-6
   = ?
   5
   
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6. m n 0 1 2 3 4 5
   10 0 1 11 111 1111 11111
   9 0 1 10 91 820 820+9^4
   8 0 1 9 73 585 585+8^4
   7 0 1 8 57 400 400+7^4
   6 0 1 7 43 256 256+6^4
   5 0 1 6 31 156 156+5^4
   4 0 1 5 21 85 341
   3 0 1 4 13 40 121
   2 0 1 3 7 15 31
   1 0 1 2 3 4 5
   6
   
   What is Pocky (Repunit) number ?
   @,,
   (Ex. @A.,,A.
   = (11).
   = 3)
   

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7. @,,-/
   = @,,
   + , (@,,A1
   = 0)
   @,,
   = I
   JA1
   ,7/
   J =
   , − 1
   − 1
   , ∈ ℕ
   lim
   ,→Q
   @,,-/
   @,,
   = , lim
   @→Q
   @-/,,
   @,,
   = 1
   7
   
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8. @,,-.
   = @,,-/
   + ,-/ = @,,-/
   + (@,,-/
   − @,,
   )
   R
   @,,-/
   = 0 ; @,,
   + 1 ; @,,-/
   @,,-.
   = −@,,
   + 1 + @,,-/
   @,,-/
   @,,-.
   =
   0 1
   − 1 +
   @,,
   @,,-/
   8
   
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9. @,,
   =
   @,,
   @,,-/
   , ℛ =
   0 1
   − 1 +
   @,,
   = ℛ,@,,A1
   @,,A1
   =
   0
   1
   9
   
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10. ℛ, =
    
    
    @,,
    =
    @,,
    @,,-/
    = ℛ,@,,A1
    =
    
    
    0
    1
    =
    
    
    @,,7/
    = ℛ,7/@,,A1
    = ℛ,ℛ7/@,,A1
    =
    
    
    −
    1
    
    0
    = −
    1
    
    
    
    10
    
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11. ℛ, =
    −@,,7/
    @,,
    −@,,
    @,,-/
    ℛ5-6 = ℛ5ℛ6
    −@,5-67/
    @,5-6
    −@,5-6
    @,5-6-/
    =
    −@,57/
    @,5
    −@,5
    @,5-/
    −@,67/
    @,6
    −@,6
    @,6-/
    @,5-6
    = @,5
    @,6-/
    − @,57/
    @,6
    11
    
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12. Change → , →
    @,5-6
    = @,5-/
    @,6
    − @,5
    @,67/
    &Y/1, /-9
    = /1,.
    /1,9
    − 10/1,/
    /1,:
    = 11 ; 1111 − 10 ; 1 ; 111 = 12221 − 1110
    = 11111
    12
    
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13. ., /-9
    = .,.
    .,9
    − 2.,/
    .,:
    = 3 ; 15 − 2 ; 1 ; 7
    = 45 − 14 = 31 = (11111).
    ., .-:
    = .,:
    .,:
    − 2.,.
    .,.
    = 7 ; 7 − 2 ; 3 ; 3
    = 49 − 18 = 31 = (11111).
    /, /-9
    = /,.
    .,9
    − /,/
    /,:
    = 2 ; 4 − 1 ; 3
    = 5 = (11111)/
    13
    
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14. "EEJUJPOUIFPSFN
    sin + = sin cos + cos sin
    cos + = cos cos − sin sin
    5-6
    = 5-/
    6
    + 5
    67/
    @,5-6
    = @,5-/
    @,6
    − @,5
    @,67/
    14
    
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15. ɾ1PDLZ OVNCFS`TBEEJUJPOUIFPSFNMJLF
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    15
    
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    m n 0 1 2 3 4 5
    10 0 1 11 111 1111 11111
    1 0 1 2 3 4 5
    

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