Need to work with number sequences? This sequence calculator handles five major sequence types: Fibonacci, Lucas, Tribonacci (recursive sequences), plus arithmetic and geometric progressions. Switch between modes to find specific terms, generate sequences, and calculate sums.
Sequence Calculator
Using the sequence calculator
Select a sequence type using the buttons at the top, enter your position (n), and see results instantly. The calculator shows the nth term, optionally displays the full sequence, and tracks how consecutive term ratios converge to mathematical constants.
Fibonacci
The Fibonacci is one of the most well known sequences. Each term is the sum of the two before it, starting with 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21...
Consecutive Fibonacci ratios converge to the golden ratio (φ ≈ 1.61803). The calculator displays this ratio so you can watch it approach φ as n increases.
For detailed formulas and examples, see our dedicated Fibonacci Sequence Calculator.
Lucas
The Lucas series follows the same rule as Fibonacci (sum of previous two), but starts with 2 and 1: 2, 1, 3, 4, 7, 11, 18, 29...
Lucas numbers are closely related to Fibonacci – every Lucas number equals the sum of adjacent Fibonacci numbers. Ratios also converge to φ.
For the Lucas-Fibonacci identity and closed-form formulas, see our Lucas Series Calculator.
Tribonacci
In the Tribonacci sequence (seriously, that's what it's called!), each term is the sum of the previous three numbers, starting with 0, 0, 1: 0, 0, 1, 1, 2, 4, 7, 13, 24, 44...
Tribonacci ratios converge to the Tribonacci constant (τ ≈ 1.83929) – larger than φ because each term incorporates more previous values.
For growth comparisons and the Tribonacci constant formula, see our Tribonacci Sequence Calculator.
Other sequence types
Beyond recursive sequences, we have dedicated tools for the two most common progression types:
Arithmetic sequences
Sequences where you add the same amount each time: 2, 5, 8, 11, 14... (adding 3). Our Arithmetic Sequence Calculator finds the nth term and sum of any arithmetic progression.
Geometric sequences
Sequences where you multiply by the same ratio each time: 2, 6, 18, 54... (multiplying by 3). Our Geometric Sequence Calculator computes nth terms, finite sums, and infinite sums when |r| < 1.
The golden ratio connection
Fibonacci and Lucas sequences both converge to the golden ratio:
\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887...This remarkable constant appears throughout mathematics, nature, and design. Our Golden Ratio Calculator helps you find golden proportions and check if dimensions form a golden ratio.
Dedicated calculators
For focused tools with detailed explanations and formulas:
