!
"#
$
% !
!
"#
%
& % '
!
! " # $!!%&
(
' )*+ ,-- -$+
!
!
"#
"
' .
/
% !
!
"#
Jn jn '
J0 = 0,
J1 = 1,
Jn = Jn−1 + 2 Jn−2 n 2
j0 = 2,
j1 = 1,
jn = jn−1 + 2 jn−2 n 2
% Jk A001045 01 & jk A014551 % 2
031 4 & & 5' % Jk 0 1 1 3 5 11 21 43 85 171
341 6 ' Jn = 2 −(−1)
3
% jk 2 1 5 7 17 31 65 127 257
511 1025 6 ' jn = 2n + (−1)n 6 & !
"#
. ' ' k ≥ 0 m ≥ 0 & '()*!&
n
n
m
jk+i & jm+k+1 − jk + 32 · (−1)k · (1 + (−1)m )
6 & & m + 1 "
i=0
!
"#
% & !
"#
7
!
"#
4
' & 4 5
& "
m i
i=0 (−1) jk+i 13 · (jk + (−1)m jk+m+1) + (−1)k (m + 1).
% ' "
& ( ' 8 .
"
.
/ 9 & :
4
#
( ("#
!
"
' ' ; . ;
01 0<1 0*1 0=1 0>1 01 0-1 %' 0+1
< % ' ' k + (k + 1) + (k + 2) + · · · + (k + n)
k n
? k(n + 1) +
(n + 1)(2k + n)
n(n + 1)
=
.
2
2
6 & ' ' &"& !
"#
6 & & &
m
i=0 jk+i & m ≥ 0 k ≥ 0 ' N∗ = {0, 1, 2, 3, 4, . . . } @&
2 A N∗∗ N∗ × N∗
+ m
i=0
(m, k) ∈ N∗∗ jk+i = jm+k+1 − jk +
3
· (−1)k · (1 + (−1)m ).
2
% & ' m
@<A
4 m = 0 & m0 jk+i jk & jk+1 − jk + 3(−1)k 6 & m @<A 2 jk = jk+1 + 3(−1)k & @+A 01
$ @<A m = r # = −1 %
r+1
jk+i =
i=0
r
jk+i + jk+r+1 =
i=0
3 k
3
(1 + r ) = jk+r+2 + 3 k+r+1 − jk + k (1 + r )
2
2
3
= jk+(r+1)+1 − jk + k (1 + r+1 ).
2
@<A m = r + 1
2 jk+r+1 − jk +
? * % & 5 , r 0 k 0 @*A
: Jr = 2 −(−1)
jr = 2r + (−1)r & ' 3
@*A 22k+2r+2
2 (J2r+3 − J2r+1 ) + 3 J2k (J2r+4 − J2r+2 )
r
=
j2k+2r+2 − 1.
r
(m, k) ∈ N∗∗ + m
j2k+2i = m + 2 J2m+1 + 3 J2m+2 J2k .
i=0
@*<A
% & ' m
4 m = 0 & m0 j2k+2i j2k & 2 + 3 J2k J1 = J2 = 1 6 & m @*<A j2k = 2 + 3 J2k & @<*A 01
$ @*<A m = r %
r+1
j2k+2i =
i=0
r
j2k+2i + j2k+2r+2 =
i=0
r + 2 J2r+1 + 3 J2r+2 J2k + j2k+2r+2 = r + 1 + 2 J2(r+1)+1 + 3 J2(r+1)+2 J2k ,
& #
? @*<A m = r + 1 = % & #
6 5 , r 0 k 0 22k+1 (J2r+4 − J2r+2 ) = j2k+2r+3 + 1.
@=A
jr = 2r + (−1)r & @=A 22k+2r+3
: Jr =
2r −(−1)r
3
(m, k) ∈ N∗∗ + -
m
j2k+1+2i = 22k+1 J2m+2 − m − 1.
i=0
@=<A
% ' m
4 m = 0 & m0 j2k+1+2i j2k+1 & 22k+1 − 1 J2 = 1 6 & m @=<A j2k+1 = 22k+1 − 1 & !
"#
$ @=<A m = r %
r+1
j2k+1+2i =
i=0
2
r
i=0
2k+1
j2k+1+2i + j2k+2r+3 =
J2r+2 − r − 1 + j2k+2r+3 = 22k+1 J2(r+1)+2 − (r + 1) − 1,
& #
< ? @=<A m = r + 1 -
+ m
(m, k) ∈ N∗∗ (−1)i jk+i =
i=0
1
· (jk + (−1)m jk+m+1 ) + (−1)k (m + 1).
3
@-A
% & ' m
4 m = 0 & m0 (−1)i jk+i jk & 13 (jk + jk+1 ) + (−1)k 6 & m @-A 2 jk = jk+1 + 3(−1)k & @+A 01
$ @-A m = r # = −1 %
r+1
i=0
i
jk+i =
r
i jk+i + r+1 jk+r+1 =
i=0
1
(jk + r jk+r+1 ) + k (r + 1) + r+1 jk+r+1 =
3
1
jk + r+1 jk+(r+1)+1 + k ((r + 1) + 1),
3
& 2 jk+r+1 = jk+r+2 − 3 k−r ?
@-A m = r + 1
>
(m, k) ∈ N∗∗ + .
m
(−1)i j2k+2i =
i=0
1
3
· (j2k + (−1)m j2k+2m+2 ) +
· (1 + (−1)m ) .
5
10
@>A
% & ' m
4 m = 0 & m0 (−1)i j2k+2i j2k & j +j 5 +3 6 & m @>A
4 j2k = j2k+2 + 3 &
& ' 4 22k + 1 = 22k+2 + 1 + 3
$ @>A m = r # = −1 %
2k
r+1
2k+2
i
j2k+2i =
i=0
r
i j2k+2i + r+1 j2k+2r+2 =
i=0
1
3
(j2k + r j2k+2r+2 ) +
(1 + r ) + r+1 j2k+2r+2
5
10
3 1 j2k + r+1 j2k+2(r+1)+2 +
1 + r+1 ,
=
5
10
& ' j2k+2r+4 = 4 j2k+2r+2 − 3 & j2k+4 = 4 j2k+2 − 3 ? @>A m = r + 1 + m
(m, k) ∈ N∗∗ (−1)i j2k+2i+1 =
i=0
1
3
· (j2k+1 + (−1)m j2k+2m+3 ) −
· (1 + (−1)m ) .
5
10
% & ' m
4 m = 0 & m0 (−1)i j2k+1+2i j2k+1 & j +j5 −3 6 & m = 0 @A 4 j2k+1 = j2k+3 −3 & & ' 4 22k+1 − 1 = 22k+3 − 1 − 3.
$ @A m = r # = −1 %
2k+1
r+1
i=0
i j2k+1+2i =
2k+3
r
i=0
i j2k+1+2i + r+1 j2k+2r+3 =
1
3
(j2k+1 + r j2k+2r+3 ) −
(1 + r ) + r+1 j2k+2r+3
5
10
1 3 j2k+1 + r+1 j2k+2(r+1)+3 −
1 + r+1 ,
=
5
10
& ' 4 j2k+2r+3 = j2k+2r+5 − 3 & 4 j2k+1 = j2k+3 − 3 ? @A m = r + 1 3 6 & "
!
"#
r 0 k 0 , -
2
3 (jk+r+1
− 1) = j2k+2r+4 − j2k+2r+2 + (−1)k+r+1 (jk+r+3 + jk+r+2 ).
r
r
: jr = 2 + (−1)
& ' 12 22k+2r − (−1)k+r 2k+r .
+ (m, k) ∈ N∗∗ m
i=0
2
jk+i
j2k+2m+2 + (−1)m+k jk+m+2 − j2k + (−1)k jk+1
.
3
% & ' m2m
4 m = 0 k& k 0 jk+i jk2 & 1 + j2k+2 +(−1) jk+23−j2k +(−1) jk+1 6 & m m+1+
3 (jk2 − 1) = j2k+2 − j2k + (−1)k (jk+2 + jk+1 )
& 3 22k + 6 (−1)k 2k $ m = r # = −1 %
r+1
2
jk+i
=
i=0
r
2
2
jk+i
+ jk+r+1
=
i=0
j2k+2r+2 + k+r jk+r+2 − j2k + k jk+1
2
+ jk+r+1
3
j2k+2(r+1)+2 + k+(r+1) jk+(r+1)+2 − j2k + k jk+1
,
= (r + 1) + 1 +
3
& #
* ? m = r + 1
r+1+
+ % & 5 & !
"#
, r 0 k 0 2
15 (j2k+2r+2
− 1) = j4k+4r+8 − j4k+4r+4 + 40 (j2k+2r+2 − j2k+2r ).
) ' 120 24k+4r+1 + 22k+2r .
m 0 k 0 + m
2
j2k+2i
=
i=0
j4k+4m+4 + 40 j2k+2m − j4k − 10 j2k
+ m − 1.
15
% & ' m
2
4 m = 0 & m0 j2k+2i
j2k2 & j
+30 j −j
− 1 6 & m 15
4k+4
2k
4k
2
15 (j2k
+ 1) = j4k+4 − j4k + 30 j2k
& 15 24k + 30 22k + 30
$ m = r %
r+1
2
j2k+2i
=
i=0
r
2
2
j2k+2i
+ j2k+2r+2
=
i=0
j4k+4r+4 + 40 j2k+2r − j4k − 10 j2k
2
+ j2k+2r+2
15
j4k+4(r+1)+4 + 40 j2k+2(r+1) − j4k − 10 j2k
= (r + 1) − 1 +
,
15
& #
= ? m = r + 1
r−1+
6 & !
"#
r 0 k 0 , .
2
15 (j2k+2r+3
− 1) = 64(j4k+4r+4 − j4k+4r ) − 80 (j2k+2r+2 − j2k+2r ).
) 240 24k+4r+2 − 22k+2r .
+ m
m 0 k 0 2
j2k+2i+1
=
i=0
64 j4k+4m − 80 j2k+2m − 4 j4k + 5 j2k+2
+ m + 2.
15
% & ' m
2
2
4 m = 0 & m0 j2k+2i+1
j2k+1
& 60 j −80 j +5 j
+ 2 6 & m 15
4k
2k
2k+2
2
15 (j2k+1
− 2) = 60 j4k − 80 j2k + 5 j2k+2
& 15 (24k+2 − 22k+2 − 1)
$ m = r %
r+1
2
j2k+2i+1
i=0
=
r
2
2
j2k+2i+1
+ j2k+2r+3
=
i=0
64 j4k+4r − 80 j2k+2r − 4 j4k + 5 j2k+2
2
+ j2k+2r+3
15
64 j4k+4(r+1) − 80 j2k+2(r+1) − 4 j4k + 5 j2k+2
= (r + 1) + 2 +
,
15
& #
- ? m = r + 1
r+2+
$ & & 7
!
"#
"
? & B 01 + 12 + 23 + 34 + · · · 01 + 23 + 45 + 67 + · · ·
12 + 34 + 56 + 78 + · · ·
, r 0 k 0 3 (jk+r+1 jk+r+2 + 1) = 8(j2k+2r+2 − j2k+2r ) − (−1)k+r (jk+r+2 + jk+r+1 ).
: jr = 2r + (−1)r & ' 3 22k+2r+3 − (−1)k+r 2k+r+1 .
+ (m, k) ∈ N∗∗ m
i=0
jk+i jk+i+1
8 j2k+2m + (−1)k+m jk+m+1 − 2 j2k + (−1)k jk
− m − 3.
3
% & ' m
4 m = 0 & k
m0 jk+i jk+i+1 jk jk+1 & 6 j2k +(−1)3 (jk+1+jk ) − 3 6 & m
3 (jk jk+1 + 3) = 6 j2k + (−1)k (jk+1 + jk )
& 3 (22k+1 + (−2)k + 2)
$ m = r %
r+1
i=0
jk+i jk+i+1 =
r
i=0
k+r
jk+i jk+i+1 + jk+r+1 jk+i+2 =
jk+r+1 − 2 j2k + (−1)k jk
− r − 3 + jk+r+1 jk+i+2
3
8 j2k+2(r+1) + (−1)k+(r+1) jk+(r+1)+1 − 2 j2k + (−1)k jk
− (r + 1) − 3,
=
3
& #
> ? m = r + 1
8 j2k+2r + (−1)
< % ' & 7
!
"#
"
&
' & 5 , r 0 k 0 15 (j2k+2r+2 j2k+2r+3 + 1) = j4k+4r+9 − j4k+4r+5 + 5 (j2k+2r+4 − j2k+2r+2 ).
r
: jr = 2r + (−1)
& ' 15 · 22k+2r+2 22k+2r+3 + 1 .
+ m
(m, k) ∈ N∗∗ j4m+4k+5 − j4k+1 j2m+2k+2 − j2k
+
− m − 1.
15
3
j2k+2i j2k+2i+1 =
i=0
% & ' m
4 m = 0 & m0 j2k+2i j2k+2i+1 j2k j2k+1 & j 15−j + j 3−j − 1 6 & m 4k+5
4k+1
2k+2
2k
15 (j2k j2k+1 + 1) = j4k+5 − j4k+1 + 5 (j2k+2 − j2k )
& 15 22k (22k+1 + 1)
$ m = r %
r+1
j2k+2i j2k+2i+1 =
i=0
r
j2k+2i j2k+2i+1 + j2k+2r+2 j2k+2r+3 =
i=0
j4k+4r+5 − j4k+1 j2k+2r+2 − j2k
+
− r − 1 + j2k+2r+2 j2k+2r+3
15
3
j4k+4(r+1)+5 − j4k+1 j2k+2(r+1)+2 − j2k
+
− (r + 1) − 1,
=
15
3
& #
? m = r + 1
* % ' & 7
!
"#
&
'
& 5 # v = 4k + 4r w = 2k + 2r
, r 0 k 0 15 (jw+3 jw+4 + 1) = jv+11 − jv+7 − 5 (jw+5 − jw+3 ).
r
: jr = 2r + (−1)
& ' 15 · 22k+2r+3 22k+2r+4 − 1 .
+ m
i=0
(m, k) ∈ N∗∗ j2k+2i+1 j2k+2i+2 =
j4m+4k+7 − j4k+3 j2m+2k+3 − j2k+1
−
− m − 1.
15
3
% & ' m
4 m = 0 & m0 j2k+2i+1 j2k+2i+2 j2k+1 j2k+2 & j 15−j − j −j3 − 1 6 & m 4k+7
4k+3
2k+3
2k+1
15 (j2k+1 j2k+2 + 1) = j4k+7 − j4k+3 − 5 (j2k+3 − j2k+1 )
& 15 22k+1 (22k+2 − 1)
$ m = r %
r+1
j2k+2i+1 j2k+2i+2 =
i=0
r
j2k+2i+1 j2k+2i+2 + j2k+2r+3 j2k+2r+4 =
i=0
j4k+4r+7 − j4k+3 j2k+2r+3 − j2k+1
−
− r − 1 + j2k+2r+3 j2k+2r+4
15
3
j4k+4(r+1)+7 − j4k+3 j2k+2(r+1)+3 − j2k+1
=
+
− (r + 1) − 1,
15
3
& #
3 ? m = r + 1
= 6 5 & 5 6 2 "
' , r 0 k 0 2
5 jk+r+1
+ 37 = j2k+2r+4 + j2k+2r+2 + 20 (−1)k+r+1(jk+r+1 − jk+r ).
) 5 22k+2r+2 + 5 (−1)k+r+1 2k+r+2 + 42.
+ -
(−1)m ·
(m, k) ∈ N∗∗ m
i=0
2
(−1)i jk+i
2 j2k+2m+2 − 37
2 j2k − 7
+ (−1)k · (4 jk+m − jk+1 ) +
.
10
10
% & ' m
4 m = 0 & jk2 & 2 j 10 −37 + (−1)k (4 jk − jk+1) +
6 & m 2k+2
m
i 2
0 (−1) jk+i
2 j2k −7
10
5 jk2 + 22 = j2k+2 + j2k + 5 (−1)k (4 jk − jk+1 )
& 5 22k + 5 (−1)k 2k+1 + 27
$ m = r # = −1 %
r+1
i=0
2
i jk+i
=
r
2
2
i jk+i
+ r+1 jk+r+1
=
i=0
2 j2k − 7
2 j2k+2r+2 − 37
2
+ k (4 jk+r − jk+1 ) +
+ r+1 jk+r+1
r
10
10
2 j2k+2(r+1)+2 − 37
2 j2k − 7
= r+1
+ k (4 jk+(r+1) − jk+1 ) +
,
10
10
& #
+ ? m = r + 1
- 6 & !
"#
& % & 5 r 0 k 0 , 2
170 j2k+2r+2
− 218 = 10(j4k+4r+8 + j4k+4r+4 ) + 34(j2k+2r+5 + j2k+2r+3 ).
: jr = 2r + (−1)r & 2720 · 24k+4r + 1360 · 22k+2r − 48
+ (−1)m ·
(m, k) ∈ N∗∗ m
i=0
2
(−1)i j2k+2i
10 j4k+4m+4 + 34 j2k+2m+3 + 109 10 j4k + 34 j2k+1 + 109
+
.
170
170
% & ' m
4 m = 0 & m
0
2
(−1)i j2k+2i
j2k2 & 10 j4k+4 + 34 j2k+3 + 109 10 j4k + 34 j2k+1 + 109
+
.
170
170
6 & m 2
170 j2k
− 218 = 10(j4k+4 + j4k ) + 34(j2k+3 + j2k+1 )
& 170 · 24k + 340 · 22k − 48.
$ m = r %
r+1
i
(−1)
2
j2k+2i
i=0
=
r
(−1)i j2k+2i + (−1)r+1 j2k+2r+2 =
i=0
10 (λ j4k+4r+4 + j4k ) + 34 (λj2k+2r+3 + j2k+1 ) + 109(λ + 1)
+ μ j2k+2r+2
170
10 (μ j4k+4(r+1)+4 + j4k ) + 34 (μ j2k+2(r+1)+3 + j2k+1 ) + 109(μ + 1)
,
=
170
& λ = (−1)r μ = (−1)r+1 = −λ #
?
m = r + 1
> 6 & !
"#
& % & 5 r 0 k 0 , 2
170 j2k+2r+3
− 218 = 10(j4k+4r+10 + j4k+4r+6 ) − 34(j2k+2r+6 + j2k+2r+4 ).
: jr = 2r + (−1)r & 10880 · 24k+4r − 2720 · 22k+2r − 48
+ .
(−1)m ·
(m, k) ∈ N∗∗ m
i=0
2
(−1)i j2k+2i+1
10 j4k+4m+6 − 34 j2k+2m+4 + 109 10 j4k+2 − 34 j2k+2 + 109
+
.
170
170
% & ' m
4 m = 0 & m
0
2
(−1)i j2k+2i+1
2
j2k+1
& 10 j4k+6 − 34 j2k+4 + 109 10 j4k+2 − 34 j2k+2 + 109
+
.
170
170
6 & m 2
170 j2k+1
− 218 = 10(j4k+6 + j4k+2 ) − 34(j2k+4 + j2k+2 )
& 680 · (24k + 22k ) − 48.
$ m = r %
r+1
2
(−1)i j2k+2i+1
=
i=0
r
(−1)i j2k+2i+1 + (−1)r+1 j2k+2r+3 =
i=0
10 (λ j4k+4r+6 + j4k+2 ) − 34 (λj2k+2r+4 + j2k+2 ) + 109(λ + 1)
+ μ j2k+2r+3
170
10 (μ j4k+4(r+1)+6 + j4k+2 ) − 34 (μ j2k+2(r+1)+4 + j2k+2 ) + 109(μ + 1)
,
=
170
& λ = (−1)r μ = (−1)r+1 = −λ #
?
m = r + 1
% & 5 r 0 k 0 , jk+r+1 jk+r+2 − 8 = j2k+2r+6 + j2k+2r+4 + 10 · (−1)k+r+1 · (jk+r+2 − jk+r+1 ).
: jr = 2r + (−1)r & 80 · 22k+2r − 20 · (−1)k+r · 2k+r − 18
+ (m, k) ∈ N∗∗ m
(−1)i jk+i jk+i+1
i=0
!" (−1)m
j2k+2 + 4
j2m+2k+4 + 4
+ (−1)k (jm+k+1 − jk ) +
.
10
10
% & ' m
4 m = 0 & 6 &
m
0
(−1)i jk+i jk+i+1
jk jk+1 & j2k+2 + 4
j2k+4 + 4
+ (−1)k (jk+1 − jk ) +
.
10
10
m 10 jk jk+1 − 8 = j2k+4 + j2k+2 + 10 (−1)k (jk+1 − jk )
& 20 · 22k + 10 · (−1)k · 2k − 18.
$ m = r %
r+1
i=0
(−1)i jk+i jk+i+1 =
r
(−1)i jk+i jk+i+1 + μ jk+r+1 jk+r+2 =
i=0
j2k+2 + 4
j2k+2r+4 + 4
λ
+ ν (jk+r+1 − jk ) +
+ μ jk+r+1 jk+r+2
10
10
j2k+2(r+1)+4 + 4
j2k+2 + 4
=μ
+ ν (jk+(r+1)+1 − jk ) +
,
10
10
& λ = (−1)r μ = (−1)r+1 = −λ ν = (−1)k #
<
? m = r + 1
3
# u = 2k + 2i v = 4k + 4r w = 2k + 2r
r 0 k 0 , -
170 jw+2 jw+3 + 218 = 10(jv+9 + jv+5 ) + 34(jw+4 + jw+2 ).
: jr = 2r + (−1)r & 5440 · 24k+4r + 680 · 22k+2r + 48
+ (−1)m ·
(m, k) ∈ N∗∗ m
i=0
(−1)i j2k+2i j2k+2i+1
10 j4k+4m+5 + 34 j2k+2m+2 − 109 10 j4k+1 + 34 j2k − 109
+
.
170
170
% & ' m
4 m = 0 & m0 (−1)i j2k+2i j2k+2i+1 j2k j2k+1 & j −109
10 j +34170j −109 + 10 j +34
6 & 170
m 4k+5
2k+2
4k+1
2k
170 j2k j2k+1 + 218 = 10 (j4k+5 + j4k+1 ) + 34 (j2k+2 + j2k )
& 340 24k + 170 22k + 48
$ m = r %
r+1
i=0
β ju ju+1 =
r
β ju ju+1 + μ jw+2 jw+3 =
i=0
10 (λ jv+5 + j4k+1 ) + 34 (λ jw+2 + j2k ) − 109(λ + 1)
+ μ jw+2 jw+3
170
10 (μ jv+9 + j4k+1 ) + 34 (μ jw+4 + j2k ) − 109(μ + 1)
=
,
170
& β = (−1)i λ = (−1)r μ = (−1)r+1 = −λ #
*
? m = r + 1
+ C & 7
!
"#
& &
' r 0 k 0 , 170 jw+3 jw+4 + 218 = 10(jv+11 + jv+7 ) − 34(jw+5 + jw+3 ).
: jr = 2r + (−1)r & 21760 · 24k+4r − 1360 · 22k+2r + 48
(m, k) ∈ N∗∗ + m
(−1)i j2k+2i+1 j2k+2i+2
i=0
!" m
(−1)
j4m+4k+7 j2m+2k+3 109
j4k+3 j2k+1 109
−
−
−
−
.
+
17
5
170
17
5
170
% & ' m
4 m = 0 & m0 (−1)i j2k+2i+1 j2k+2i+2 j2k+1 j2k+2 & 10 j −34170j −109 + 10 j −34170j −109 6 & m 4k+7
2k+3
4k+3
2k+1
170 j2k+1 j2k+2 + 218 = 10 (j4k+7 + j4k+3 ) − 34 (j2k+3 + j2k+1 )
& 1360 · 24k − 340 · 22k + 48
$ m = r %
r+1
β ju+1 ju+2 =
i=0
r
β ju+1 ju+2 + μ jw+3 jw+4 =
i=0
10 (λ jv+7 + j4k+3 ) − 34 (λ jw+3 + j2k+1 ) − 109(λ + 1)
+ μ jw+3 jw+4
170
10 (μ jv+11 + j4k+3 ) − 34 (μ jw+5 + j2k+1 ) − 109(μ + 1)
=
,
170
& β = (−1)i λ = (−1)r μ = (−1)r+1 = −λ #
=
? m = r + 1
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