Find but do not solve a recurrence relation for an the numbe

Find -but do not solve- a recurrence relation for a_n, the number of sequences of 2s, 5s, and 7s whose terms sum to n with the condition of no subsequence of 257. Give appropriate initial conditions and find a₉.

Solution

The c_i are any constants and the base of the exponents are the roots to the characteristic can be verified by induction.

If the characteristic has a multiple root, this step is modified slightly.

If r is a root of multiplicity m, use (c₁rⁿ + c₂nrⁿ + c₃n²rⁿ + ... + c_mn^m-1rⁿ) instead of simply (c₁rⁿ).

For example, the sequence starting 5, 0, -4, 16, 144, 640, 2240, ... satisfies the recursive relationship    a_n = 6a_n-1 - 12a_n-2 + 8a_n-3.

The number of sequences of 2s, 5s, and 7s whose terms sum to n with the condition of no subsequence of 257. Give appropriate initial conditions and find a₉.

The characteristic polynomial has a triple root of 2 and the closed form formula

                                       a_n = 5*2ⁿ - 7*n*2ⁿ + 2*n²*2ⁿ.

a₉₌5(2⁹)-7(9)(2⁹)+2(9²)(2⁹)=53248