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A109065 Numerator of the fraction due in month n of the total interest for a one-year installment loan based on the Rule of 78s (each denominator is 78). 0
12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 (list; graph; refs; listen; history; text; internal format)
The method is called the "Rule of 78s" or the "Sum-of-the-Digits" method because 1 + 2 + 3 + ... + 12 = 78 (=A000217(12)). This sequence is the first twelve terms of A022968, which is row 12 of triangle A004736. If, instead, the loan is, say, for six months or two years, A004736's row 6 (6,5,4,3,2,1) or row 24 (24,23,...,1) is applied and the denominator becomes 21 (=A000217(6)) or 300 (=A000217(24)), respectively. (The 78 is always the number appearing in the name of the general method.) A disadvantage of the Rule of 78s for the borrower (in contrast with the "actuarial method" where the fractions are the same for each month; e.g., 1/12 for each month of a one-year loan) is that if the loan is repaid early, the heavier weighting of interest in the earlier months causes a higher effective interest rate sometimes known as a prepayment penalty. The Rule of 78s dates back to the 1920's. It was adopted because of ease of use, but its current legality varies by state and the loan's term.
Tanya Khovanova, Non Recursions
a(n) = 13 - n (1 <= n <= 12).
a(1) = 12 because 12/78 (=2/13) is the fraction of the total precalculated loan interest considered accrued in the first month and payable in the first monthly payment of a Rule of 78s loan with one-year term.
Cf. A022968 (12-n, n>=0), A004736 (triangle T(n, k) = n-k, n>=1, 0<=k<n), A000217 (triangular numbers).
Sequence in context: A233513 A175042 A255134 * A022968 A023454 A055123
Rick L. Shepherd, Jun 17 2005

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