These are the sums of elements on successive diagonals of a left-justified Pascal's triangle beginning in the leftmost column and moving in steps of up and 1 right. The case equals the usual Fibonacci
number. These numbers satisfy the identities

(2)

(3)

(4)

(5)

(Bicknell-Johnson and Spears 1996). For the special case ,

(6)

Bicknell-Johnson and Spears (1996) give many further identities.

Horadam (1965) defined the generalized Fibonacci numbers as , where , , , and are integers, , , and for . They satisfy the identities

(7)

(8)

(9)

(10)

where

(11)

(12)

(Dujella 1996). The final above result is due to Morgado (1987) and is called the
morgado identity.

Another generalization of the Fibonacci numbers is denoted . Given and , define the generalized Fibonacci number by for ,

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74-81, 1971.Bicknell-Johnson, M. and Spears, C. P. "Classes
of Identities for the Generalized Fibonacci Numbers for Matrices with Constant Valued Determinants."
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Fibonacci Numbers and the Problem of Diophantus." Fib. Quart.34,
164-175, 1996.Horadam, A. F. "Generating Functions for Powers
of a Certain Generalized Sequence of Numbers." Duke Math. J.32,
437-446, 1965.Horadam, A. F. "Generalization of a Result of
Morgado." Portugaliae Math.44, 131-136, 1987a.Horadam,
A. F. and Shannon, A. G. "Generalization of Identities of Catalan
and Others." Portugaliae Math.44, 137-148, 1987b.Morgado,
J. "Note on Some Results of A. F. Horadam and A. G. Shannon Concerning a Catalan's
Identity on Fibonacci Numbers." Portugaliae Math.44, 243-252,
1987.