Atom
November 11, 2019
120

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

## Atom

November 11, 2019

## Transcript

1. 1PDLZ OVNFSJDBMTFRVFODF
3FQVOJU OVNCFS`T BEEJUJPOUIFPSFN
# &OHMJTI4FNJOBS

/BHBPLB 6OJWFSTJUZ PG 5FDIOPMPHZ
"UPN :PTIJ[BXB

2. \$POUFOUT

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(2) What is Pocky (Repunit) number ?

1PDLZ OVNCFS`T CBTJDUIFPSFN

1PDLZ OVNCFS`TBEEJUJPOUIPSFN

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3. 5PEBZ /PWJT1PDLZ EBZ
#ZUIFXBZ
%PZPVLOPXBEEJUJPOUIFPSFN
sin + = sin cos + cos sin
cos + = cos cos − sin sin
3

*OUSPEVDUJPO

4. 'JCPOBDDJOVNCFS
,-.
= ,-/
+ ,
, 1
= 0, /
= 1, .
= 1, ⋯
'JCPOBDDJOVNCFS`TBEEJUJPOUIFPSFN
5-6
= 5-/
6
+ 5
67/
&Y8
= /-9
= /
:
+ .
9
= 1 ; 2 + 1 ; 3 = 5
4

*OUSPEVDUJPO

5. 1PDLZ(Repunit) OVNCFS
@A/1,,
= ,
%FDJNBMTZTUFN

1
= 0, /
= 1, .
= 11, :
= 111, 9
= 1111,
*TUIFSF1PDLZ OVNCFS`TBEEJUJPOUIFPSFN
5-6
= ?
5

*OUSPEVDUJPO

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)

7. @,,-/
= @,,
+ , (@,,A1
= 0)
@,,
= I
JA1
,7/
J =
, − 1
− 1
, ∈ ℕ
lim
,→Q
@,,-/
@,,
= , lim
@→Q
@-/,,
@,,
= 1
7

1PDLZ OVNCFS`T CBTJDUIFPSFN

8. @,,-.
= @,,-/
+ ,-/ = @,,-/
+ (@,,-/
− @,,
)
R
@,,-/
= 0 ; @,,
+ 1 ; @,,-/
@,,-.
= −@,,
+ 1 + @,,-/
@,,-/
@,,-.
=
0 1
− 1 +
@,,
@,,-/
8

1PDLZ OVNCFS`TBEEJUJPOUIPSFN

9. @,,
=
@,,
@,,-/
, ℛ =
0 1
− 1 +
@,,
= ℛ,@,,A1
@,,A1
=
0
1
9

1PDLZ OVNCFS`TBEEJUJPOUIPSFN

10. ℛ, =

@,,
=
@,,
@,,-/
= ℛ,@,,A1
=

0
1
=

@,,7/
= ℛ,7/@,,A1
= ℛ,ℛ7/@,,A1
=

1

0
= −
1

10

1PDLZ OVNCFS`TBEEJUJPOUIPSFN

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

1PDLZ OVNCFS`TBEEJUJPOUIPSFN

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

1PDLZ OVNCFS`TBEEJUJPOUIPSFN

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

1PDLZ OVNCFS`TBEEJUJPOUIPSFN

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

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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