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

1PDLZ OVNFSJDBMTFRVFODF 3FQVOJU OVNCFS`T BEEJUJPOUIFPSFN # &OHMJTI4FNJOBS /BHBPLB 6OJWFSTJUZ
PG 5FDIOPMPHZ "UPN :PTIJ[BXB

$POUFOUT *OUSPEVDUJPO (2) What is Pocky (Repunit) number ?
1PDLZ OVNCFS`T CBTJDUIFPSFN 1PDLZ OVNCFS`TBEEJUJPOUIPSFN $PODMVTJPO

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

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

1PDLZ(Repunit) OVNCFS @A/1,, = , %FDJNBMTZTUFN 1 = 0, /
= 1, . = 11, : = 111, 9 = 1111, *TUIFSF1PDLZ OVNCFS`TBEEJUJPOUIFPSFN 5-6 = ? 5 *OUSPEVDUJPO

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)

@,,-/ = @,, + , (@,,A1 = 0) @,, =
I JA1 ,7/ J = , − 1 − 1 , ∈ ℕ lim ,→Q @,,-/ @,, = , lim @→Q @-/,, @,, = 1 7 1PDLZ OVNCFS`T CBTJDUIFPSFN

@,,-. = @,,-/ + ,-/ = @,,-/ + (@,,-/ −
@,, ) R @,,-/ = 0 ; @,, + 1 ; @,,-/ @,,-. = −@,, + 1 + @,,-/ @,,-/ @,,-. = 0 1 − 1 + @,, @,,-/ 8 1PDLZ OVNCFS`TBEEJUJPOUIPSFN

@,, = @,, @,,-/ , ℛ = 0 1 −
1 + @,, = ℛ,@,,A1 @,,A1 = 0 1 9 1PDLZ OVNCFS`TBEEJUJPOUIPSFN

ℛ, = @,, = @,, @,,-/ = ℛ,@,,A1 = 0
1 = @,,7/ = ℛ,7/@,,A1 = ℛ,ℛ7/@,,A1 = − 1 0 = − 1 10 1PDLZ OVNCFS`TBEEJUJPOUIPSFN

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

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

., /-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

"EEJUJPOUIFPSFN sin + = sin cos + cos sin cos
+ = cos cos − sin sin 5-6 = 5-/ 6 + 5 67/ @,5-6 = @,5-/ @,6 − @,5 @,67/ 14 $PODMVTJPO

ɾ1PDLZ OVNCFS`TBEEJUJPOUIFPSFNMJLF DPTJOFGVODUJPOJOUSJHPOPNFUSJDBEEJUJPOUIFPSFNT ɾ1PDLZ OVNCFS`TBEEJUJPOUIFPSFNJT FYUFOTJPOPG OBUVSBMOVNCFS`TUIBU 15 $PODMVTJPO
m n 0 1 2 3 4 5 10 0 1 11 111 1111 11111 1 0 1 2 3 4 5

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

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

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1PDLZ OVNFSJDBMTFRVFODF 3FQVOJU OVNCFS`T BEEJUJPOUIFPSFN # &OHMJTI4FNJOBS /BHBPLB 6OJWFSTJUZ

$POUFOUT *OUSPEVDUJPO (2) What is Pocky (Repunit) number ?

5PEBZ /PWJT1PDLZ EBZ #ZUIFXBZ %PZPVLOPXBEEJUJPOUIFPSFN sin + = sin

'JCPOBDDJOVNCFS ,-. = ,-/ + , , 1 = 0,

1PDLZ(Repunit) OVNCFS @A/1,, = , %FDJNBMTZTUFN 1 = 0, /

m n 0 1 2 3 4 5 10 0

@,,-/ = @,, + , (@,,A1 = 0) @,, =

@,,-. = @,,-/ + ,-/ = @,,-/ + (@,,-/ −

@,, = @,, @,,-/ , ℛ = 0 1 −

ℛ, = @,, = @,, @,,-/ = ℛ,@,,A1 = 0

ℛ, = −@,,7/ @,, −@,, @,,-/ ℛ5-6 = ℛ5ℛ6 −@,5-67/

Change → , → @,5-6 = @,5-/ @,6 − @,5

., /-9 = .,. .,9 − 2.,/ .,: = 3

"EEJUJPOUIFPSFN sin + = sin cos + cos sin cos

ɾ1PDLZ OVNCFS`TBEEJUJPOUIFPSFNMJLF DPTJOFGVODUJPOJOUSJHPOPNFUSJDBEEJUJPOUIFPSFNT ɾ1PDLZ OVNCFS`TBEEJUJPOUIFPSFNJT FYUFOTJPOPG OBUVSBMOVNCFS`TUIBU 15 $PODMVTJPO