## 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?

math grad - Interest: Number theory
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### 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.

• benvitalis says:

Note that the values of n are x^2 – 2