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Special Sums of Generalized Fibonacci Numbers Jim Cullen 
Phi =  Sqrt( 5 ) + 1 2 
phi =  Sqrt( 5 )  1 2 
F( z ) =  Phi^{z}  ( phi )^{z} Sqrt( 5 ) 
Fib( z ) =  Phi^{z}  cos( Pi * z ) * phi^{z} Sqrt( 5 ) 
G( a, b, z ) =  a * F( z  1 ) + b * F( z ) 




inf  
F( i + d ) r^{( i + k )}  =  G( 1, r, d + 1 ) r^{k} ^{.} ( r^{2}  r  1 )  
i = 1 
inf  
F( i + d ) r^{( i + k )}  =  G( 1, r, d ) r^{( k  1 )} ^{.} (r^{2}  r  1 )  
i = 0 
inf  
F( i + d ) r^{( i + k )}  =  G( 1, r, d + m ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )  
i = m 


inf  
G( a, b, i ) r^{i}  = a +  a + br r^{2}  r  1  
i = 0 
inf  
G( a, b, i ) + k r^{i}  
i = m 
r^{( 1  m )} r  1 
( 1  r ) r^{2}  r  1  ^{.} G[ ( r  1 ) ^{.} b  ( r  2 ) ^{.} a, b + ( r  1 ) ^{.} a, m + 1 ] 
inf  
G( a, b, i ) + k r^{i}  = r^{( 1  m )} ^{.}  [  k r  1  +  r ^{.} G( a, b, m ) + G( a, b, m  1 ) r^{2}  r  1  ]  
i = m 
inf  
G( a, b, i ) r^{i}  =  r ^{.} G( a, b, m ) + G( a, b, m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
G( a, b, i ) r^{i}  =  G( b + ar  a, br + a, m ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
F( i ) r^{i}  = r^{( 1  m )} ^{.}  [  G( r  1, 1, m + 1 ) r^{2}  r  1  ]  
i = m 
inf  
L( i ) r^{i}  = r^{( 1  m )} ^{.}  [  G( 3  r, 2 ^{.} r  1, m + 1 ) r^{2}  r  1  ]  
i = m 
inf  
F( i ) r^{i}  =  r ^{.} F( m ) + F( m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
L( i ) r^{i}  =  r ^{.} L( m ) + L( m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
G( a, b, i ) r^{i}  =  r ^{.} G( a, b, m ) + G( a, b, m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
F( i + d ) r^{( i + k )}  =  G( 1, r, d + m ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
G( a, b, i ) r^{i}  =  r ^{.} G( a, b, m ) + G( a, b, m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
G( a, b, i + d ) r^{( i + k )}  
i = m 
inf  inf  inf  
F( i + d ) r^{( i + k )}  = a ^{.}  F( i + d  1 ) r^{( i + k )}  + b ^{.}  F( i + d ) r^{( i + k )}  
i = m  i = m  i = m 
a ^{.} G( 1, r, d + m  1 ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )  +  b ^{.} G( 1, r, d + m ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 ) 
inf  
G( a, b, i + d ) r^{( i + k )}  =  a ^{.} G( 1, r, d + m  1 ) + b ^{.} G( 1, r, d + m ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )  
i = m 
inf  
G( a, b, i + d ) r^{( i + k )}  =  G( b + ar  a, br + a, d + m ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )  
i = m 




inf  
i ^{.} G( a, b, i ) r^{i}  =  r ^{.} ( b r^{2} + 2 a r + b a ) ( r^{2}  r  1 )^{2}  
i = 0;1 
inf  
i ^{.} F( i + d ) r^{i}  =  G( r^{3} + r, r^{3} + 2r^{2}, d ) ( r^{2}  r  1 )^{2}  
i = 0;1 
inf  
i ^{.} F( i + d ) r^{i}  =  r ^{.} G( r^{2} + 1, r^{2} + 2r, d ) ( r^{2}  r  1 )^{2}  
i = 0;1 
inf  
i ^{.} G( a, b, i ) r^{i}  
i = m 
inf  
i ^{.} G( a, b, i ) r^{i}  =  p( r ) ^{.} G( a, b, m ) + q( r ) ^{.} G( a, b, m + 1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )^{2}  
i = m 
inf  
i ^{.} G( a, b, i + d ) r^{( i + k )}  =  p( r ) ^{.} G( a, b, d + m ) + q( r ) ^{.} G( a, b, d + m + 1 ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )^{2}  
i = m 
inf  
i ^{.} G( a, b, i + d ) r^{( i + k )}  =  p( r ) ^{.} G( a, b, d + m ) + q( r ) ^{.} G( a, b, d + m  1 ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )^{2}  
i = m 
inf  
i ^{.} G( a, b, i ) r^{i}  =  p( r ) ^{.} G( a, b, m ) + q( r ) ^{.} G( a, b, m + 1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )^{2}  
i = m  
with  
p( r ) = m r^{3}  2 m r^{2} + 2 r + m  1  
and  
q( r ) = ( m + 1 ) r^{2}  m r  m + 1 
inf  
i ^{.} G( a, b, i ) r^{i}  =  p( r ) ^{.} G( a, b, m ) + q( r ) ^{.} G( a, b, m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )^{2}  
i = m  
with  
p( r ) = m r^{3} + ( 1  m ) r^{2} + ( 2  m ) r  
and  
q( r ) = ( m + 1 ) r^{2}  m r  m + 1 
inf  
i ^{.} G( a, b, i + d ) r^{( i + k )}  =  p( r ) ^{.} G( a, b, d + m ) + q( r ) ^{.} G( a, b, d + m  1 ) r^{( m + k  1 )} ^{.} ( r^{2}  r  1 )^{2}  
i = m  
with  
p( r ) = m r^{3} + ( 1  m ) r^{2} + ( 2  m ) r  
and  
q( r ) = ( m + 1 ) r^{2}  m r  m + 1 
inf  
i^{k} ^{.} G( a, b, i ) r^{i}  =  p( r ) ^{.} G( a, b, m ) + q( r ) ^{.} G( a, b, m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )^{k+1}  
i = m 
inf  
i^{2} ^{.} G( a, b, i ) r^{i}  =  p( r ) ^{.} G( a, b, m ) + q( r ) ^{.} G( a, b, m  1 ) r^{( m  1 )} ^{.} ( r^{2}  r  1 )^{3}  
i = m  
with  
p( r ) = m^{2}r^{5} + ( 2m^{2} + 2m + 1 )r^{4} + ( m^{2} + 2m + 5 )r^{3} + ( 2m^{2}  6m + 3 )r^{2} + ( 2  m )^{2} r  
and  
q( r ) = ( m + 1 )^{2}r^{4} + ( 2m^{2}  2m + 1 )r^{3} + ( 6  m^{2} )r^{2} + ( 2m^{2}  2m  1 )r + ( m  1 )^{2} 