C(n,k), C(n,k+1) and C(n,k+2) in A.P.

 
 

Find all positive integers   n   and   k   such that the three binomial coefficients

\dbinom{ \;n \;}{ \;k \;} ,   \dbinom{ \;n \;}{ \;k + 1 \;}   and   \dbinom{ \;n \;}{ \;k + 2 \;}

are in arithmetic progression
 
 

Is it possible for three consecutive binomial coefficients to be in geometric progression?
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

2 Responses to C(n,k), C(n,k+1) and C(n,k+2) in A.P.

  1. paul says:

    There are 2 linear recurrence that satisfies this problem

    LinearRecurrence[{3,-3,1},{7,14,23},20] &
    LinearRecurrence[{3,-3,1},{1,4,8},20]

    which gives these sequences
    n = {7,14,23,34,47,62,79,98,119,142,167,194,223,254,287,322,359,398,439,482}
    k = {1,4,8,13,19,26,34,43,53,64,76,89,103,118,134,151,169,188,208,229}

    and in pairs for {n,k} so

    {7,1}, {14, 4}.. etc

    There is also a negative version starting with the second term of k thus
    {7,4}, {14 ,8}, {23,13}.. etc

    So there are an infinite number of solutions.

    I can’t find any that are in GP.

    Paul.

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