Compute Lucas terms and sequence totals accurately. Review recurrence steps, ratios, downloadable tables, and charts. Get clean outputs for study, checking, teaching, and sharing.
Use the responsive grid below. It shows three columns on large screens, two on smaller screens, and one on mobile.
Recurrence relation: L0 = 2, L1 = 1, and Ln = Ln-1 + Ln-2 for n ≥ 2.
Closed form: Ln = φn + ψn, where φ = (1 + √5) / 2 and ψ = (1 - √5) / 2.
Range total: The calculator adds each displayed Lucas value across the selected range to produce a running total and average.
Ratio check: For exact mode, the ratio Ln / Ln-1 approaches the golden ratio as n grows larger.
This sample shows the first eleven Lucas numbers with cumulative totals.
| Index | Lucas Value | Cumulative Sum |
|---|---|---|
| 0 | 2 | 2 |
| 1 | 1 | 3 |
| 2 | 3 | 6 |
| 3 | 4 | 10 |
| 4 | 7 | 17 |
| 5 | 11 | 28 |
| 6 | 18 | 46 |
| 7 | 29 | 75 |
| 8 | 47 | 122 |
| 9 | 76 | 198 |
| 10 | 123 | 321 |
A Lucas number belongs to a recurrence sequence related to Fibonacci numbers. It starts with 2 and 1, then each new term equals the sum of the previous two terms.
Both sequences follow the same recurrence rule. The difference is the starting values. Fibonacci starts with 0 and 1, while Lucas starts with 2 and 1.
As the index grows, the dominant part of the recurrence behaves like powers of the golden ratio. That makes consecutive-term ratios settle near 1.6180339 in exact mode.
Use modulo when you want reduced values for pattern work, coding tasks, or number theory checks. It is also helpful when you want larger indexes without exact large integer growth.
This page uses standard integer arithmetic for exact values. Keeping the exact index limit at 88 avoids overflow and preserves reliable outputs for the shown summaries and tables.
The running total adds each displayed Lucas value from the selected range, row by row. It helps you inspect accumulated growth and compare sequence windows quickly.
The CSV includes summary metrics and the full sequence table. The PDF captures the visible result section, including cards, chart, and calculated table.
Yes. The page shows the computed term, a range table, a chart, and the main formulas. That makes it useful for demonstrations, checking answers, and reviewing sequence behavior.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.