Recursive Sequence Calculator

How it is calculated

You supply two starting values and three constants, and the tool builds each new term from the two before it: a(n) = p × a(n−1) + q × a(n−2) + r. The first term listed is a0, the second is a1, the third is p × a1 + q × a0 + r, and so on. Number of terms N says how many terms to list, counting a0 as the first. N must be from 1 to 50; a fractional N is cut to its whole part, and N = 1 shows only a0. Coefficients and seeds may be negative or decimal. The tool returns no result if any term becomes non-finite or larger than about 9 × 10¹⁵ in size, because past that point digits would no longer be exact. The list always uses plain dot decimals with no thousands separators, so it can be copied; the last term is shown separately in the page's number format. With p = q = 1, r = 0 and seeds 0 and 1 you get the Fibonacci numbers.

Common mistakes

• Expecting the count to start from a1 or to leave out a0. The list always starts with a0, so 8 terms ends at the term built from index 7.
• Treating the constant r as a multiplier. It is added once at every step, not multiplied by an earlier term.
• Swapping the coefficients. p multiplies the term just before, q the one before that; swapping them gives a different sequence.
• Asking for more than 50 terms, or for 0 terms. Both give no result.
• Picking coefficients that grow so fast that a term passes about 9 × 10¹⁵, for example p = q = 2 over many terms. The tool then shows nothing rather than inexact digits; ask for fewer terms.