%N 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).
%C 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.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/NonRecursions.html">Non Recursions</a>
%F a(n) = 13 - n (1 <= n <= 12).
%e 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.
%Y Cf. A022968 (12-n, n>=0), A004736 (triangle T(n, k) = n-k, n>=1, 0<=k<n), A000217 (triangular numbers).
%A _Rick L. Shepherd_, Jun 17 2005