26 sept 2016 · Anti-Fibonacci Numbers: A Formula Notes by Thomas Zaslavsky Binghamton University (SUNY) Binghamton, New York 13902-6000
Abstract A Pisot number is a real algebraic integer ? > 1 such that all of its Galois conjugates, other than ? itself, lie inside the unit circle
The positive integer n is said to be an anti-tau number if gcd(n, ?(n)) = 1 Part (b) of the above theorem shows that the anti-tau numbers are unlike
SYMMETRIC AND ANTI SYMMETRIC KRONECKER SQUARES AND INTERTWINING NUMBERS OF INDUCED REPRESENTATIONS OF FINITE GROUPS * By GEoRGE W MACKEY Introduction
4 juil 2022 · Find all integers n ? 3 for which there exist real numbers a1,a2 is an anti-Pascal triangle with four rows which contains every integer
24 juil 2020 · groups were studied by co-author Young, in [11], where the anti-van der Waerden numbers were connected to the order of the group
The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós For given graphs G and H the
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ABAC AD=AE BD
CE ABAC FG
DEFG n3 a1;a2;:::;an aiai+1+ 1 =ai+2 i= 1;2;:::;n n 1 10 4 26
571
83109
2018
11 + 2 ++ 2018
(x;y) x;y2 f1;:::;20g
400
p5 K K a1a2 N nN a1 a2+a2 a3++an 1 an+an a1 (an) ABCDABCD=BCDA X ABCD \XAB=\XCD\XBC=\XDA: \BXA+\DXC= 180
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ABC DE
ABAC AD=AE BD
CE ABAC FG
DEFG
ǥ AXFDAEGY
XY XFkAB M
ÔXFdAB NA
BCFMN G DEXY XF=AD=AE=Y G ÔXFdY G
ÔMFÔNG MNkFG
MNDE \A Ƕ
ǥ bcfga
d a=
2f+a+b abf
2 b a=f ab f e a=g ac g AD=AE e a d a 2 =c b)(g2 ac)2 (f2 ab)2=g2c f2b )bc(bg2 cf2)a2=g2f4c f2g4b=f2g2(f2c g2b) )bca2= (fg)2) fg a 2 =bc: fg a X AX?FG FG \A \A
ǥ ` \A DF
G0 FG0?`E0 AC GE0=GC ě AD=AE0 E=E0
FG0?` \FBD=1
2C+x\GCA=1
2B+x x FGkMNMN \FAB=1
2C x\GAC=1
2B x
R
BF= 2R
1 2C x :
BD= 2BF
1 2C+x = 4R 1 2C+x 1 2C x
DA=AB BD= 2RC 4R
1 2C+x 1 2C x = 2R C 2 1 2C+x 1 2C x = 2R[C (C 2x)] = 2R2x: AE0= 2R2x
FG0kDE G=G0
ǥ FDGE J
K A BCDE FGJ K 4BFD 4JAD FB=FD AJ=AD
4CGE 4KAEGC=GE AK=AE
AK=AE=AD=AJ;
DEJK A
]KED=]KJD=]KJF=]KGF; Ƕ KJ GK ACK ABC AK=AD=AE(AB;AC)
K AB J AC K6=J:
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n3 a1;a2;:::;an aiai+1+ 1 =ai+2 i= 1;2;:::;n n 3jn ( 1; 1;2; 1; 1;2;:::)
ǥ aiai+1ai+2
aiai+1ai+2= [ai+2 1]ai+2=a2i+2 ai+2 =ai[ai+3 1] =aiai+3 ai: a2i+2 ai+2=aiai+3 ai X i a2i+2=X i aiai+3()X (ai ai+3)2= 0: 3
1 x2+ 1 =x 3jn
ǥ an
+ >1 ++ a1= 0 a2= 0a3>0a4>0 ++ + + a1= x <0 a2= y <0 a3= 1 +xy >1 a4= 1 y(1 +xy)<0 a5= 1 + (1 +xy)(1 y(1 +xy))<1: a6 a5= (1 +a5a4) (1 +a3a4) =a4(a5 a3)>0 a5>1> a3 ++ + + + +::: a1a2+ 1 =a3<0< a4= 1 +a2a3)a1< a3 a2>0 a1< a3< a5< ::: + + + +
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1 10 4 26
571
83109
2018
11 + 2 ++ 2018
n= 2018N= 1 + 2 ++n d d d=a b d!a A 1 + 2 ++n
N= 1 + 2 ++n f1;2;:::;ng B
A BC DXY XY B B C C D bn/2 1c n+1n+2 1;2;:::;n A!B D (n+ 1) + (n+ 2) ++ (n+ (bn/2 1c))>1 + 2 ++n n2018
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(x;y) x;y2 f1;:::;20g
400
p5 K K K= 100 100
200
1
2200 = 100
100
44 44 2 664
1 2 3 4
3 4 1 2
2 1 4 3
4 3 2 1
3 775
4 1/4
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a1a2 N nN a1 a2+a2 a3++an 1 an+an a1 (an)
S(n) =an+1 an
a1+an an+1 n > N p p p n > N p(an)< p(an+1) p(an+1) =p(a1) p(an) =p(an+1) p(an)> p(an+1) p(an+1)p(a1) ěp p(aN+1)p(aN+2) p(aN+1)p(aN+2)::: K > N p(aK)< p(aK+1) =p(aK+2) ==p(a1) dzǴ p(a1) p(a1)>0 ν (a1)n > N pa1 p(an) pja1 an+1jan n p an2Zan2Zp p xn p(xn) xn xn n
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an/a1 n=N+ 1;N+ 2;::: 1 1
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ABCDABCD=BCDA X ABCD \XAB=\XCD\XBC=\XDA: \BXA+\DXC= 180
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ABCD=BCDA
ě X
ABCD \D=\A\A=\B
DABCABCD:
DABC ABCD X X X ABCD \BXA+\DXC= 180 XX
ǥ dz Ǵ
AB AD=CB CD BA BC=DA DC BD !ACAC A
C !BDBD
PABCD P 1 ABCD 1
A0B0=PA
ABPB=PC
BCPB=1
B0C0 A0B0=B0C0 B0C0=C0D0=D0A0 ]XAB=]XCD ]XAB=]XAP+]PAB=]PX0A0+]A0B0P ]XCD=]XCP+]PCD=]PX0C0+]C0D0P ]A0X0C0=]A0B0P+]PD0C0=](A0B0;B0P) +](PD0;C0D0) =](A0B0;B0P) +](PD0;A0B0) =]D0PB0:
A0B0kC0D0
]B0X0D0=]A0PC0: ]AXB=]AXP+]PXB=]PA0X0+]X0B0P ]CXD=]CXP+]PXD=]PC0X0+]X0DP0 ]AXB+]CXD= (]PA0X0+]X0B0P) + (]PC0X0+]X0D0P) = ]A0X0P+]X0PA0+ ]PX0B0+]B0PX0 ]C0X0P+]X0PC0+ ]PX0D0+]D0PX0 =]PX0A0+]BX0P+]PX0C0+]D0X0P +]A0PX0+]X0PB0+]C0PX0+]X0PD0 =]A0PB0+]C0PD0+]B0X0C+]D0X0A ]A0PB0+]C0PD0+]B0X0C+]D0X0A= 0: A B CDX YQ P
Q P Y (BQC)
(AQD) ]AXC= ]DPB=]BQD=]BQY+]Y QD=]BCY+]Y AD =](BC;CY) +](Y A;AD) =]Y CA= ]AY C:
XACY XBDY
ěX6=Y AB < AC Q !BD T BA(QBC) QB=QD=QT=pQAQC \BAD <180 (QBCY)
ABCD ABCD
X ]DXA=]DXY+]Y XA=]DBY+]Y CA =](DB;Y B) +](CY;CA) =]CY B+ 90 =]CQB+ 90= ]APB+ 90: ]BXC=]DPC+ 90: X X Y